JC-NRLF 


$B    Mlfi    3D1 


THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 


\ 


HIGHER  MATHEMATICS. 


A  TEXT-BOOK  FOR 


CLASSICAL  AND  ENGINEERING  COLLEGES. 


EDITED   BY 


•       MANSFIELD    MERRIMAN, 

Professor  of  Givil  Engineering  in  Lehigh  University, 


ROBERT   S.  WOODWARD, 

Professor  of  Mechanics  in  Columbia  University. 


FIRST    EDITION 
FIRST    THOUSAND. 


NEW  YORK: 

JOHN   WILEY  &   SONS. 

London:    CHAPMAN  &  HALL,  Limited. 

1896. 


MAIH  UB.. 


Copyright,  1896, 

BY 

MANSFIELD  MERRIMAN 

AND 

ROBERT  S.  WOODWARD. 


ROBERT  DRUMMOND,    ELECTROTVPER   AND   PRINTER,   NEW   YORK, 


PREFACE. 


In  the  early  part  of  this  century  it  was  possible  for  an  in- 
dustrious  student  to  acquire  a  comprehensive  if  not  minute 
knowledge  of  the  entire  realm  of  mathematical  science.  The 
more  eminent  minds  of  that  time,  Hke  Lagrange,  Laplace,  and 
Gauss,  were  about  equally  familiar  with  all  branches  of  pure 
and  applied  mathematics.  Since  that  epoch  the  tendency  has 
been  constantly  towards  specialization ;  and  additions  to  pure 
theory  along  with  extensions  of  apphcations  have  been  made 
with  increasing  rapidity,  until  now  the  mere  quantity  of  in- 
formation available  presents  a  formidable  obstacle  to  the  simul- 
taneous attainment  of  the  breadth  and  depth  of  knowledge 
which  characterized  the  mathematician  of  a  generation  ago. 
It  would  appear,  however,  that .  this  obstacle  is  due  to  the 
bewildering  mass  of  details  rather  than  to  any  considerable 
increase  in  the  number  of  fundamental  principles.  Hence  the 
student  who  seeks  to  gain  a  comprehensive  view  of  the  mathe- 
matics of  the  present  day  needs  most  of  all  that  sort  of  guid- 
ance which  fixes  his  attention  on  essentials  and  prevents  him 
from  wasting  valuable  time  and  energy  in  the  pursuit  of  non- 
essentials. 

During  the  past  twenty  years  a  marked  change  of  opinion 
has  occurred  as  to  the  aims  and  methods  of  mathematical 
instruction.  The  old  ideas  that  mathematical  studies  should 
be  pursued  to  discipHne  the  mind,  and  that  such  studies  were 
ended  when  an  elementary  course  in  the  calculus  had  been 
covered,  have  for  the  most  part  disappeared.  In  our  best 
classical  and    engineering  colleges   the   elementary  course  in 

iii 

:30  3SLH'b 


IV  PREFACE. 

calculus  is  now  given  in  the  sophomore  year,  while  lectures  and 
seminary  work  in  pure  mathematics  are  continued  during  the 
junior  and  senior  years.  It  is  with  the  hope  of  meeting  the 
existing  demand  for  a  suitable  text  to  be  used  in  such  upper- 
class  work  that  the  editors  enlisted  the  cooperation  of  the 
authors  in  the  task  of  bringing  together  the  chapters  of  this 
book.  It  was  the  intention  of  the  editors  to  include  a  chapter 
on  elliptic  integrals  and  functions ;  much  to  their  regret,  how- 
ever, it  was  found  impracticable  to  obtain  the  manuscript  in 
time  for  publication.  Notwithstanding  this  omission  the  volume 
contains  about  one-fifth  more  matter  than  was  originally  con- 
templated. 

Each  chapter,  so  far  as  it  goes,  is  complete  in  itself,  and  is 
intended  primarily  to  give  a  clear  idea  of  the  leading  prin- 
ciples of  the  subject  treated.  While  the  authors  have  been 
guided  by  general  instructions  issued  by  the  editors,  each  has 
been  free  to  follow  his  own  plan  of  treatment.  It  will  be  found 
that  certain  chapters  adopt  the  formal  method  usual  in  text- 
books, while  others  employ  what  may  be  called  the  historical 
and  intuitive  method.  A  glance  at  the  table  of  contents  will 
show  that  the  chapters  of  the  work  present  a  considerable 
variety  of  subjects,  thus  affording  teachers  and  students  an 
opportunity  to  select  such  topics  as  maybe  suited  to  their  time 
and  tastes.  Numerous  problems  are  given  for  solution,  numer- 
ical examples  of  the  application  of  theory  to  physical  science 
are  freely  introduced,  and  the  foot-notes  set  forth  much  sug- 
gestive matter  of  a  historical  and  critical  nature. 

The  Editors. 

June  30,  1896, 


CONTENTS. 


Chapter  I.     THE  SOLUTION  OF  EQUATIONS. 

By  Mansfield  Merriman, 
Professor  of  Civil  Engineering  in  Lehigh  University. 

Article  i.  Introduction Page  r 

2.  Graphic  Solutions   3 

3.  The  Regula  Falsi 5 

4.  Newton's  Approximation  Rule 6 

5.  Separation  of  the  Roots 8 

6.  Numerical  Algebraic  Equations 10 

7.  Transcendental  Equations 13 

8.  Algebraic  Solutions .' 15 

9.  The  Cubic  Equation 17 

10.  The  Quartic  Equation ig 

11.  Quintic  Equations 21 

12.  Trigonometric  Solutions 24 

13.  Real  Roots  by  Series 27 

14.  Computation  of  All  Roots 28 

15.  Conclusion 31 

Chapter  II.     DETERMINANTS. 

By  Laenas  Gifford  Weld, 
Professor  of  Mathematics  in  the  State  University  of  Iowa. 

Article  I.   Introduction 33 

2.  Permutations 34 

3.  Interchange  of  Two  Elements 35 

4.  Positive  and  Negative  Permutations 36 

5.  The  Determinant  Array 3^J 

6.  Determinant  as  Function  of  n^  Elements 37 

7.  Examples  of  Determinants 38 

8.  Notations 39 

9.  Second  and  Third  Orders 40 

10.  Interchange  of  Rows  and  Columns -12 

11.  Interchange  of  Two  Parallel  Lines -:3 

12.  Two  Identical  Parallel  Lines -^3 

V 


VI  CONTENTS. 

Article  13.  Multiplying  by  a  Factor Page  44 

14.  A  Line  of  Polynomial  Elements. ..  .c 44 

15.  Composition  of  Parallel  Lines 45 

16.  Binomial  Factors 46 

17.  Co-factors ;  Minors 47 

18.  Development  in  Terms  of  Co-factors 49 

19.  The  Zero  Formulas 5^ 

20.  Cauchy's  Method  of  Development 52 

21.  Differentiation  of  Determinants 54 

22.  Raising  the  Order 55 

23.  Solution  of  Linear  Equations 56 

24.  Consistence  of  Linear  Systems 58 

25.  The  Matrix 60 

26.  Homogeneous  Linear  Systems 60 

27.  Co-factors  in  a  Zero  Determinant 62 

28.  Sylvester's  Method  of  Elimination 63 

29.  The  Multiplication  Theorem 65 

30.  Product  of  Two  Rectangular  Arrays 68 

31.  Reciprocal  Determinants 69 

Chapter  IH.     PROJECTIVE  GEOMETRY. 

By  George  Bruce  Halsted, 
Professor  of  Mathematics  in  the  University  of  Texas. 

Article  i.  The  Elements  and  Primal  Forms 70 

2.  Projecting  and  Cutting 72 

3.  Elements  at  Infinity 72 

4.  Correlation  and  Duality 74 

5.  Polystims  and  Polygrams 74 

6.  Harmonic  Elements 77 

7.  Projectivity 80 

8.  Curves  of  the  Second  Degree 82 

9.  Pole  and  Polar 87 

10.  Involution 88 

Projective  Conic  Ranges gi 

Center  and  Diameter 94 

13.  Plane  and  Point  Duality 96 

14.  Ruled  Quadric  Surfaces g8 

15.  Cross  Ratio 104 

Chapter  IV.     HYPERBOLIC  FUNCTIONS. 

By  James  McMahon, 
Associate  Professor  of  Mathematics  in  Cornell  University. 

Article  i.   Correspondence  of  Points  on  Conies 107 

2.  Areas  of  Corresponding  Triangles 109 

3.  Areas  of  Corresponding  Sectors 109 

4.  Characteristic  Ratios  of  Sectorial  Measures no- 


il. 

12. 


CONTENTS.  Vii 

Article  5.   Ratios  expressed  as  Triangle-measures Page  no 

6.  Functional  Relations  for  Ellipse m 

7.  Functional  Relations  for  Hyperbola m 

8.  Relations  between  Hyperbolic  Functions 112 

9.  Variations  of  the  Hyperbolic  Functions 114 

10.  Anti-hyperbolic  Functions 116 

11.  Functions  of  Sums  and  Differences 116 

12.  Conversion  Formulas 118 

13.  Limiting  Ratios 119 

14.  Derivatives  of  Hyperbolic  Functions 120 

15.  Derivatives  of  Anti-hyperbolic  Functions 122 

16.  Expansion  of   Hyperbolic  Functions 123 

17.  Exponential  expressions 124 

18.  Expansion  of  Anti-functions 125 

19.  Logarithmic  Expression  of  Anti-functions 127 

20.  The  Gudermanian  Function 128 

21.  Circular  Functions  of  Gudermanian 128 

22.  Gudermanian  Angle. 129 

23.  Derivatives  of  Gudermanian  and  Inverse 130 

24.  Series  for  Gudermanian  and  its  Inverse 131 

25.  Graphs  of  Hyperbolic  Functions 132 

26.  Elementary  Integrals 135 

27.  Functions  of  Complex  Numbers 138 

28.  Addition  Theorems  for  Complexes 140 

29.  Functions  of  Pure  Imaginaries 141 

30.  Functions  of  ^  -|-  i^  in  the  Form  X-\-  iV 143 

31.  The  Catenary I45 

32.  The  Catenary  of  Uniform  Strength I47 

33.  The  Elastic  Catenary 148 

34.  The  Tractory I49 

35.  The  Loxodrome 150 

36.  Combined  Flexure  and  Tension 151 

37.  Alternating  Currents i53 

38.  Miscellaneous  Applications 158 

39.  Explanation  of  Tables 160 

Chapter  V.     HARMONIC  FUNCTIONS. 

By  William  E.  Byerly, 
Professor  of  Mathematics  in  Harvard  University, 

Article  I.   History  and  Description 169 

2.  Homogeneous  Linear  Differential  Equations 172 

3.  Problem  in  Trigonometric  Series I74 

4.  Problem  in  Zonal  Harmonics ^77 

5.  Problem  in  Bessel's  Functions 183 

6.  The  Sine  Series ^^^ 

7.  The  Cosine  Series ^92 

8.  Fourier's  Series ^94 


Tin  CONTENTS. 

Article  9.   Extension  of  Fourier's  Series Page  196 

ID.  Dirichlet's  Conditions 198 

11.  Applications  of  Trigonometric  Series 200 

12.  Properties  of  Zonal  Harmonics 202 

13.  Problems  in  Zonal  Harmonics 205 

14.  Additional  Forms 207 

15.  Development  in  terms  of  Zonal  Harmonics 208 

16.  Formulas  for  Development - 209 

17.  Formulas  in  Zonal  Harmonics 212 

18.  Spherical  Harmonics.    213 

19.  Bessel's  Functions.     Properties 213 

20.  Applications  of  Bessel's  Functions 215 

21.  Development  in  Terms  of  Bessel's  Functions 217 

22.  Problems  in  Bessel's  Functions 220 

23.  Bessel's  Functions  of  Higher  Order 221 

24.  Lame's  Functions 221 

25.  Tables 222 

Chapter  VI.     FUNCTIONS  OF  A  COMPLEX  VARIABLE. 

By  Thomas  S.  Fiske, 
Adjunct  Professor  of  Mathematics  in  Columbia  University. 

Article  i.  Definition  of  Function 226 

2.  Representation  of  Complex  Variable 227 

3.  Absolute  Convergence 228 

4.  Elementary  Functions 229 

5.  Continuity  of  Functions 230 

6.  Graphical  Representation  of  Functions 232 

7.  Derivatives 233 

8.  Conf ormal  Representation .' 236 

9.  Examples  of  Conformal  Representation 238 

10.  Conformal  Representation  of  a  Sphere 244 

It.  Conjugate  Functions 245 

12.  Application  to  Fluid  Motion 246 

13.  Critical  Points 250 

14.  Point  at  Infinity 256 

15.  Integral  of  a  Function 257 

16.  Reduction  of  Complex  Integrals  to  Real 26: 

17.  Cauchy's  Theorem 262 

18.  Application  of  Cauchy's  Theorem 264 

19.  Theorems  on  Curvilinear  Integrals 267 

20.  Taylor's  Series 269 

21.  Laurent's  Series 271 

22.  Fourier's  Series 273 

23.  Uniform  Convergence 274 

24.  One-valued  Functions  with  Critical  Points 278 

25.  Residues 282 

26.  Integral  of  a  One-valued  Function 284 


CONTENTS.  IX 

Article  27.  Weierstrass's  Theorem Page  287 

28.  Mittag-Leffler's  Theorem 292 

29.  Critical  Lines  and  Regions 298 

30.  Functions  having  n  Values 300 


Chapter  VII.     DIFFERENTIAL  EQUATIONS. 

By  W.  WooLSEY  Johnson, 
Professor  of  Mathematics  in  United  States  Naval  Academy. 

Article  l.   Equations  of  First  Order  and  Degree 303 

2.  Geometrical  Representation 305 

3.  Primitive  of  a  Differential  Equation 307 

4.  Exact  Differential  Equations 308 

5.  Homogeneous   Equation 311 

6.  The  Linear  Equation 312^ 

7.  First  Order  and  Second  Degree 314 

8.  Singular  Solutions 317 

9.  Singular  Solution  from  the  Complete  Integral 320 

10.  Solution  by  Differentiation 322 

11.  Geometric  Applications;  Trajectories 325 

12.  Simultaneous  Differential  Equations 327 

13.  Equations  of  the  Second  Order 330 

14.  The  First  Two  Integrals .333 

15.  Linear  Equations 336 

16*  Linear  Equations  with  Constant  Coefficients 338 

17.  Homogeneous  Linear  Equations 342 

18.  Solutions  in  Infinite  Series 344 

19.  Systems  of  Differential  Equations 349 

20.  First  Order  and  Degree  with  Three  Variables 352 

21.  Partial  Differential  Equations  of  First  Order  and  Degree...  355 

22.  Complete  and  General  Integrals 359 

23.  Complete  Integral  for  Special  Forms 362 

24.  Partial  Equations  of  Second  Order 365 

25.  Linear  Partial  Differential  Equations 3^8 

Chapter  VIII.     GRASSMANN'S  SPACE  ANALYSIS. 

By  Edward  W.  Hyde, 
Professor  of  Mathematics  in  the  University  of  Cincinnati. 

Article  i.   Explanations  and  Definitions 374 

2.  Sum  and  Difference  of  Two  Points 375 

3.  Sum  of  Two  Weighted  Points 37^ 

4.  Sum  of  any  Number  of  Points •  381 

5.  Reference  Systems 3^6 

6.  Naturs  of  Geometric  Multiplication 39^ 

7.  Planimetric  Products 392 

8.  The  Complement 399 


X  CONTENTS. 

Article  9.   Equations  of  Condition  and  Formulas Page  405 

10.  Stereometric  Products 410 

11.  The  Complement  in  Solid  Space 416 

12.  Addition  of  Sects  in  Solid  Space 419 

Chapter  IX.     VECTOR  ANALYSIS  and  QUATERNIONS. 

By  Alexander  Macfarlane, 
Lecturer  in  Electrical  Engineering  in  Lehigh  University. 

Article  i.  Introduction 425 

2.  Addition  of  Coplanar  Vectors 426 

3.  Products  of  Coplanar  Vectors 432 

4.  Coaxial  Quaternions 430 

5.  Addition  of  Vectors  in  Space 443 

6.  Product  of  Two  Vectors 444 

7.  Product  of  Three  Vectors 440 

8.  Composition  of  Located  Quantities 453 

9.  Spherical   Trigonometry acj 

10.   Composition  of  Rotations 463 

Chapter  X.     PROBABILITY  and  THEORY  OF  ERRORS. 

By  Robert  S.  Woodward, 
Professor  of  Mechanics  in  Columbia  University. 

Article  i.  Introduction 467 

2.  Permutations 471 

3.  Combinations 473 

4.  Direct  Probabilities 476 

5.  Probability  of  Concurrent  Events 479 

6.  Bernouilli's  Theorem 482 

7.  Inverse  Probabilities 484 

8.  Probabilities  of  Future  Events „ 487 

9.  Theory  of  Errors 490 

10.  Laws  of  Error 491 

11.  Typical  Errors  of  a  System 493 

12.  Laws  of  Resultant  Error 494 

13.  Errors  of  Interpolated  Values 497 

14.  Statistical  Test  of  Theory 504 

Chapter  XI.     HISTORY  OF  MODERN  MATHEMATICS. 

By  David  Eugene  Smith, 
Professor  of  Mathematics  in  Michigan  State  Normal  School. 

Article  i.  Introduction 508 

2.  Theory  of  Numbers ,    511 

3.  Irrational  and  Transcendent  Numbers 513 

4.  Complex  Numbers 515 

5.  Quaternions  and  Ausdehnungslehre 517 


CONTENTS.                                                    ^  XI 

Article  6.  Theory  of  Equations Page  519 

7.  Substitutions  and  Groups 524 

8.  Determinants 526 

9.  Quantics 52S 

10.  Calculus 531 

11.  Differential  Equations 535 

12.  Vifinite  Series 539 

13.  Theory  of  Functions 543 

14.  Probabilities  and  Least  Squares 550 

15.  Analytic  Geometry 552 

16.  Modern  Geometry 558 

17.  Trigonometry  and  Elementary  Geometry 563 

18.  Non-Euclidean  Geometry 565 

19.  Bibliography 568 

Index 571 


HIGHER    MATHEMATICS 


Chapter  I. 

THE   SOLUTION   OF    EQUATIONS. 

By  Mansfield  Merriman, 
Professor  of  Civil  Engineering  in  Lehigh  University. 

Art.  1.     Introduction. 

In  this  Chapter  will  be  presented  a  brief  outline  of  methods, 
not  commonly  found  in  text-books,  for  the  solution  of  an 
equation  containing  one  unknown  quantity.  Graphic,  numeric, 
and  algebraic  solutions  will  be  given  by  which  the  real  roots 
of  both  algebraic  and  transcendental  equations  may  be  ob- 
tained, together  with  historical  information  and  theoretic 
discussions. 

An  algebraic  equation  is  one  that  involves  only  the  opera- 
tions of  arithmetic.     It  is  to  be  first  freed  from  radicals  so  as 
to  make  the  exponents  of  the  unknown  quantity  all  integers; 
the  degree  of  the  equation  is  then  indicated  by  the  highest  ex- 
ponent of  the  unknown  quantity.    The  algebraic  solution  of  an 
algebraic  equation  is  the  expression  of  its  roots  in   terms  of 
the  literal  coefificients ;  this  is  possible,  in  general,  only  for  linear, 
quadratic,  cubic,  and  quartic  equations,  that  is,  for  equations 
of  the  first,  second,  third,   and  fourth  degrees.     A  numerical 
equation  is  an  algebraic  equation  having  all  its  coefificients  real 
numbers,  either  positive  or  negative.     For  the   four  degrees 


2  THE    SOLUTION    OF    EQUATIONS.  [Chap.  I. 

above  mentioned  the  roots  of  numerical  equations  may  be 
computed  from  the  formulas  for  the  algebraic  solutions,  unless 
they  fall  under  the  so-called  irreducible  case  wherein  real 
quantities  are  expressed  in  imaginary  forms. 

An  algebraic  equation  of  the  n^'^  degree  may  be  written 
with  all  its  terms  transposed  to  the  first  member,  thus: 

x""  +  a^x""'  +  a^x''-''  +  .  .  .  +  a^-,x  -\-  a^  —  O, 

and,  for  brevity,  the  first  member  will  be  called  f{x)  and  the 
equation  be  referred  to  as/(;t).=^  O.  The  roots  of  this  equa- 
tion are  the  values  of  x  which  satisfy  it,  that  is,  those  values  of 
X  that  reduce /(;ir)  to  o.  When  all  the  coefficients  a^,  a^, .  ..a^ 
are  real,  as  will  always  be  supposed  to  be  the  case,  Sturm's 
theorem  gives  the  number  of  real  roots,  provided  they  are  un- 
equal, as  also  the  number  of  real  roots  lying  between  two 
assumed  values  of  x,  while  Horner's  method  furnishes  a  con- 
venient process  for  obtaining  the  values  of  the  roots  to  any 
required  degree  of  precision. 

A  transcendental  equation  is  one  involving  the  operations 
of  trigonometry  or  of  logarithms,  as,  for  example,  x  -{-  cos;ir  =:  o, 
or  d^ -\- xlf  =  O.  No  general  method  for  the  literal  solution 
of  these  equations  exists  ;  but  when  all  known  quantities  are 
expressed  as  real  numbers,  the  real  roots  niay  be  located  and 
computed  by  tentative  methods.  Here  also  the  equation  may 
be  designated  as/(;tr)  =  o,  and  the  discussions  in  Arts.  2-5  will 
apply  equally  well  to  both  algebraic  and  transcendental  forms. 
The  methods  to  be  given  are  thus,  in  a  sense,  more  valuable 
than  Sturm's  theorem  and  Horner's  process,  although  for 
algebraic  equations  they  may  be  somewhat  longer.  It  should 
be  remembered,  however,  that  algebraic  equations  higher  than 
the  fourth  degree  do  not  often  occur  in  physical  problems,  and 
that  the  value  of  a  method  of  solution  is  to  be  measured  not 
merely  by  the  rapidity  of  computation,  but  also  by  the  ease 
with  which  it  can  be  kept  in  mind  and  applied. 

Prob.  I.  Reduce  the  equation  {a  -\-  x)^ -\-  (a  —  x)l  =  2b  to  an 
equation  having  the  exponents  of  the  unknown  quantity  all  integers. 


Art.  2.] 


GRAPHIC    SOLUTIONS. 


Art.  2.    Graphic  Solutions. 

Approximate  values  of  the  real  roots  of  two  simultaneous 
algebraic  equations  may  be  found  by  the  methods  of  plane 
analytic  geometry  when  the  coefficients  are  numerically 
expressed.     For  example,  let  the  given  equations  be 

x''  -{-y  =  a\  x"  —  bx  —  y""  —  cy, 

the  first  representing  a  circle  and  the  second  a  hyperbola. 
Drawing  two  rectangular  axes  OX  and  OY,  the  circle  is  de- 
scribed from  O  with  the  radius  a.  The  coordinates  of  the 
center  of  the  hyperbola  are  found  to  be  OA  =  ^b  and  AC  ^^c, 
while  its  diameter  BD  =^y/ b"  —  c^^  from  which  the  tv/o 
branches  may  be  described. 
The  intersections  of  the  circle 
with  the  hyperbola  give  the 
real  values  of  x  and  y.  If 
^^  =  I,  <^  =  4,  and  ^  =  3,  there 
are  but  two  real  values  for  x 
and  two  real  values  for  y, 
since  the  circle  intersects  but 
one  branch  of  the  hyperbola  ; 
here   Om    is  the  positive  and 

Op  the  negative  value  of  x,  while  mn  is  the  positive  and  pq 
the  negative  value  of  7.  When  the  radius  a  is  so  large  that 
the  circle  intersects  both  branches  of  the  hyperbola  there  are 
four  real  values  of  both  x  and  y. 

By  a  similar  method  approximate  values  of  the  real  roots  cf 
an  algebraic  equation  containing  but  one  unknown  quantity  may 
be  graphically  found.  For  instance,  let  the  cubic  equation 
x-"  J^ax  —  b  —  o  h^  required  to  be  solved.*  This  may  be 
written  as  the  two  simultaneous  equations 

y=iX^,  y  =  —  ax  +  b, 

*See  Proceedings  of  the  Engineers'  Club  of  Philadelphia,  1884,  Vcl.  IV, 
pp.  47-49 


J-} 

C       (p       1 

fp  0 

JJ 

Vlx 

mj         2 

i 

4  THE    SOLUTION    OF    EQUATIONS.  [Chap.  I. 

and  the  graph  of  each  being  plotted,  the  abscissas  of  their 
points  of  intersection  give  the  real  roots  of  the  cubic.     The 

curve  y  =^  x^  should  be  plotted  upon 
/  cross-section  paper  by  the  help  of  a 

^.  /  table  of  cubes ;    then   OB  is  laid  off 

.___^^^><p  equal  to  b,  and  OC  equal  to  a/b,  tak- 

^  ^\C  ing  care  to  observe  the  signs  of  a  and 
b.  The  Hne  joining  B  and  C  cuts 
the  curve  at  /,  and  hence  gp  is  the 
real  root  of  x^  -\-  ax  —  b  =^  o.  If  the 
cubic  equation  have  three  real  roots  the  straight  Hne  BC  will 
intersect  the  curve  in  three  points. 

Some  algebraic  equations  of  higher  degrees  may  be  graphic- 
ally solved  in  a  similar  manner.  For  the  quartic  equation 
2"  -|-  Az^  -\-  Bz  —  C  =  o/\X.  is  best  to  put  z  =  A^x,  and  thus- 
reduce  it  to  the  form  x* -\- x"" -\- bx  —  c  =  o\  then  the  two 
equations  to  be  plotted  are 

y  =  x'-\-x\        y=—bx  +  c, 

the  first  of  which  may  be  drawn  once  for  all  upon  cross-section 
paper,  while  the  straight  line  represented  by  the  second  may 
be  drawn  for  each  particular  case,  as  described  above.* 

This  method  is  also  applicable  to  many  transcendental  equa- 
tions ;  thus  for  the  equation  Ax  —  Bsmx  =^0  it  is  best  to- 
write  ax  —  s\r\x  —  o\  then  y  =  sin;r  is  readily  plotted  by  help 
of  a  table  of  sines,  while  y  :=  ax  is  a  straight  line  passing 
through  the  origin.  In  the  same  way  a^  —  x^  =  o  gives  the 
curve  represented  by  y  =  a''  and  the  parabola  represented  by 
y  =  x^f  the  intersections  of  which  determine  the  real  roots  of 
the  given  equation. 

Prob.  2.  Devise  a  graphic  solution  for  finding  approximate 
values  of  the  real  roots  of  the  equation  x''-{-  ax^-\-  bx^-\-  ex  -\-  d  =  o. 

Prob.  3.  Determine  graphically  the  number  and  the  approximate 
values  of  the  real  roots  of  the  equation  arc  x  —  8  sin  :*:  =  o. 
(Ans. — Six  real  roots,  x  =  ±  159°?  ±  43o°>  and  ±  456°.) 

*  For  an  extension  of  this  method  to  the  determination  of  imaginary  roots,: 
see  Phillips  and  Beebe's  Graphic  Algebra,  New  York,  18&2. 


Art.  3.]  the  regula  falsi.  5 

Art.  3.    The  Regula  Falsi. 

One  of  the  oldest  methods  for  computing  the  real  root  of 
an  equation  is  the  rule  known  as  ''  regula  falsi,"  often  called 
the  method  of  double  position.*  It  depends  upon  the  princi- 
ple that  if  two  numbers  x^  and  x^  be  substituted  in  the  expres- 
sion/(;r),  and  if  one  of  these  renders /(;r)  positive  and  the  other 
renders  it  negative,  then  at  least  one  real  root  of  the  equation 
f{x)  =  o  lies  between  x^  and  x^ .  Let  the  figure  represent  a 
part  of  the  real  graph  of  the  equation  j  =/{x).  The  point  X, 
where  the  curve  crosses  the  axis  of  abscissas,  gives  a  real  root 
OX  of  the  equation  /{x)  =  o.  Let  OA  and  OB  be  inferior  and 
superior  limits  of  the  root  OX  which  are  determined  either  by- 
trial  or  by  the  method  of  Art.  5. 
Let  Aa  and  Bd  be  the  values  of 
/(x)  corresponding  to  these  limits.  q  ^ 
Join  ad,  then  the  intersection  C  of 
the  straight  hne  ad  with  the  axis 
OB  gives  an  approximate  value 
OC  for  the  root.  Now  compute 
Cc  and  join  ac,  then  the  intersection  D  gives  a  value  OD  which 
is  closer  still  to  the  root  OX. 

Let  x^  and  x^  be  the  assumed  values  OA  and  OB,  and  let 
/(x,)  3ind/{x^)  be  the  corresponding  values  of/(;ir)  represented 
by  Aa  and  Bd,  these  values  being  with  contrary  signs.  Then 
from  the  similar  triangle  AaC  and  BdC  the  abscissa  OC  is 

By  a  second  application  of  the  rule  to  x,  and  x,,  another  value 
X,  is  computed,  and  by  continuing  the  process  the  value  of  x 
can  be  obtained  to  any  required  degree  of  precision. 

As  an  example  let  /[x)  =z  x"  +  S^"  +  7  =  o.  Here  it  may 
be  found  by  trial  that  a  real  root  lies  between  -2  and  —  1.8. 

*  This  originated  in  India,  and  its  first  publication  in  Europe  was  by  Abra- 
ham ben  Esra,  in  1130.  See  Matthiesen,  Grundzuge  der  antiken  und  moder- 
nen  Algebra  der  litteralen  Gleichungen,  Leipzig,  1878. 


6 


THE    SOLUTION    OF    EQUATIONS. 


[Chap.  L 


For^,  =  -  2j{x^  =  -  5.  andfor;r,  =  -  i.8,/(a::,)  =  +4-304; 
then  by  the  regula  falsi  there  is  found  jt-g  =  —  1.90  nearly. 
Again,  for  x^—  —  1.90,  f{x^  —  -\-  0.290,  and  these  combined 
with  x^  and  f{x^  give  x^-=^  —  1. 906,  which  is  correct  to  the 
third  decimal. 

As  a  second  example  let  /(>)  =  arc;tr  —  sin;ir  —  0.5  =  o. 
Here  a  graphic  solution  shows  that  there  is  but  one  real  root, 
and  that  the  value  of  it  lies  between  85°  and  86°.  For  x^^=^  85°,, 
f{x^  =  —  0.01266,  and  for  x^  =  ^6°,f{x^  =  -\-  0.00342  ;  then 
by  the  rule  x^  —  85"  44^  which  gives /(;ir3)  =  —  0.00090.  Again, 
combining  the  values  for  x^  and  x^  there  is  found  x^  =  85°  47'. 
which  gives  /{x^)  =  —  0.00009.  Lastly,  combining  the  values 
for  x^  and  x^  there  is  found  x^  =  Sy  47'. 4,  <v'hich  is  as  close  an 
approximation  as  can  be  made  with  five-place  tables. 

In  the  application  of  this  method  it  is  to  be  observed  that 
the  signs  of  the  values  of  x  and  /{x)  are  to  be  carefully  re- 
garded, and  also  that  the  values  of  f{x)  to  be  combined  in  one 
operation  should  have  opposite  signs.  For  the  quickest 
approximation  the  values  of /(;ir)  to  be  selected  should  be  those 
having  the  smallest  numerical  values. 

Prob.  4.  Compute  by  the  regula  falsi  the  real  roots  of  :r^  — 0.25  =  0. 
Also  those  of  x"^  -\-  sin  2X  =  o. 


Art.  4.    Newton's  Approximation  Rule. 

Another  useful  method  for  approximating  to  the  value  of 
the  real  root  of  an  equation  is  that  devised  by  Newton  in  1666.^ 

If  j;  =/{^)  be  the  equation  of  a 
curve,  OX  in  the  figure  represents  a 
real  root  of  the  equation  /(x)  =  o. 
Let  OA  be  an  approximate  value  of 
OX,  and  Aa  the  corresponding  value 
oi/{x).  At  a  let  aB  be  drawn  tangent 
to  the  curve;  then  OB  is  another  approximate  value  of  OX, 

*  See  Analysis  per  equationes  numero  terminorum  infinitas.  p.  269,  Vol.  I 
of  Horsely's  edition  of  Newton's  works  (London,  1779),  where  the  method  is 
given  in  a  somewhat  different  form. 


Art.  4.]  newton's  approximation  rule.  7 

Let  Bb  be  the  value  of  f{x)  corresponding  to  OB,  and  at  b 
let  the  tangent  bC  be  drawn ;  then  OC  is  a  closer  approxima- 
tion to  OX,  and  thus  the  process  may  be  continued. 

Let/(^)  be  the  first  derivative  o{  f[x)\  oxJ\x)  —  df{x)/dx. 
For  X  =  x,-=^  OA  in  the  figure,  the  value  of  f{x^  is  the  ordi- 
nate Aa,  and  the  value  of  f\x^  is  the  tangent  of  the  angle 
aBA  ;  this  tangent  is  also  Aa/AB.  Hence  AB  =/i.^,)//'(-^i). 
and  accordingly  OB  and  OC  are  found  by 

which  is  Newton's  approximation  rule.  By  a  third  application 
to  x^  the  closer  value  x^  is  found,  and  the  process  may  be  con- 
tinued to  any  degree  of  precision  required. 

For  example,  let  f{x)  =  ;ir'  +  5;r'  -}-  7  =  O-  The  first  deriv- 
ative is/'(;ir)  =  ^x*'  -\-  \Qx.  Here  it  may  be  found  by  trial  that 
—  2  is  an  approximate  value  of  the  real  root.  For  ;ir,  =  —  2 
f{x^  =1  —  5,  and/'(;ir,)  =  6o,  whence  by  the  rule  x^-=^  —  1.92. 
Now  for  JIT,  =  —  1.92  are  found  fix^  =  —  0.6599  and 
f{x^  =  29.052,  whence  by  the  rule  x^^=  —  1.906,  which  is 
correct  to  the  third  decimal. 

As  a  second  example  let  fix)  =  ;r'' +  4sin;ir  =  o.  Here 
the  first  derivative  is /"^(;i;)  =  2;ir -|-4Cos;r.  An  approximate 
value  of  X  found  either  by  trial  or  by  a  graphic  solution  is 
;ir=  — 1.94,  corresponding  to  about  —  iii°09^  For  ;irj  =  — 1.94, 
f(x^  =z  0.03304  and  f\x^  =  —  5.323,  whence  by  the  rule 
jr,  =  —  1.934.  By  a  second  application  x^^=  —  1.9328,  which 
corresponds  to  an  angle  of  —  110°  54^'. 

In  the  application  of  Newton's  rule  it  is  best  that  the 
assumed  value  of  x^  should  be  such  as  to  render /(;ri)  as  small 
as  possible,  and  also/'(;ir,)  as  large  as  possible.  The  method 
will  fail  if  the  curve  has  a  maximum  or  minimum  between  a 
and  b.  It  is  seen  that  Newton's  rule,  like  the  regula  falsi, 
applies  equally  well  to  both  transcendental  and  algebraic  equa- 
tions, and  moreover  that  the  rule  itself  is  readily  kept  in  mind 
by  help  of  the  diagram. 


8  THE    SOLUTION    OF    EQUATIONS.  [ChAP.  I. 

Prob.  5.  Compute  by  Newton's  rule  the  real  roots  of  the  alge- 
braic equation  x*  —  jx  -{-  6  =  o.  Also  the  real  roots  of  the  trans- 
cendental equation  sin  x  +  arc  ^  —  2  =  0. 


Art.  5.     Separation  of  the  Roots. 

The  roots  of  an  equation  are  of  two  kinds,  real  roots  and 
imaginary  roots.  Equal  real  roots  may  be  regarded  as  a  spe- 
cial class,  which  lie  at  the  limit  between  the  real  and  the  imagi- 
nary. If  an  equation  has/  equal  roots  of  one  value  and  g  equal 
roots  of  another  value,  then  its  first  derivative  equation  has 
p  —  \  roots  of  the  first  value  and  q  —  i  roots  of  the  second 
value,  and  thus  all  the  equal  roots  are  contained  in  a  factor 
common  to  both  primitive  and  derivative.  Equal  roots  may 
hence  always  be  readily  detected  and  removed  from  the  given 
equation.  For  instance,  let  x"  —  gx"  -\-  /^x  -[-12  --  O,  of  which 
the  derivative  equation  is  /^x^  — i8;ir-|-4  =  o;  as  x  —  2  is  a 
factor  of  these  two  equations,  two  of  the  roots  of  the  primitive 
equation  are  -|-  2. 

The  problem  of  determining  the  number  of  the  real  and 
imaginary  roots  of  an  algebraic  equation  is  completely  solved 
by  Sturm's  theorem.  If,  then,  two  values  be  assigned  to  x  the 
number  of  real  roots  between  those  limits  is  found  by  the  same 
theorem,  and  thus  by  a  sufficient  number  of  assumptions  limits 
may  be  found  for  each  real  root.  As  Sturm's  theorem  is  known 
to  all  who  read  these  pages,  no  applications  of  it  will  be  here 
given,  but  instead  an  older  method  due  to  Hudde  will  be 
presented  which  has  the  merit  of  giving  a  comprehensive  view 
of  the  subject,  and  which  moreover  applies  to  transcendental 
as  well  as  to  algebraic  equations.* 

If  any  equation  y  =^  f{x)  be  plotted  with  values  of  x  as 
abscissas  and  values  of  j  as  ordinates,  a  real  graph  is  obtained 
whose  intersections  with  the  axis  OX  give  the  real  roots  of  the 

*  Devised  by  Hudde  in  1659  ^"^  published  by  Rolle  in  1690.  See  CEuvres 
de  Lagrange,  Vol.  VIII,  p.  190. 


Art.  5.] 


SEPARATION    OF    THE    ROOTS. 


equaf  ion  /(jtr)  =  o.  Thus  in  the  figure  the  three  points  marked 
Xg'v/e  three  values  OX  (or  three  real  roots.  The  curve  which 
repi?;sentsj/  =/{^)  has  points  of  maxima  and  minima  marked 
Ay  *nd  inflection  points  marked  B.     Now  let  the  first  deriva- 


tive equation  dyldx=f\x)  be  formed  and  be  plotted  in  the 
same  manner  on  the  axis  O'  X' .  The  condition  f'{x)=  o  gives 
the  abscissas  of  the  points  A^  and  thus  the  real  roots  0' X  give 
limits  separating  the  real  roots  o{  f(x)  =  o.  To  ascertain  if  a 
real  root  OX  lies  between  two  values  of  O'X^  these  two  values 
are  to  be  substituted  'm/(x):  if  the  signs  of/(;i;)  are  unlike  in 
the  two  cases,  a  real  root  of  /(;ir)  =  o  Hes  between  the  two 
limits;  if  the  signs  are  the  same,  a  real  root  does  not  lie  between 
those  limits. 

In  like  manner  if  the  second  derivative  equation,  that  is, 
.d''yldx''  =  f"(x\  be  plotted  on  O" X" ,  the  intersections  give 
limits  which  separate  the  real  roots  of  f\x)=.  o.  It  is  also 
seen  that  the  roots  of  the  second  derivative  equation  are  the 
abscissas  of  the  points  of  inflection  of  the  curve  j^  =i  f{x). 

To  illustrate  this  method  let  the  given  equation  be  the 
quintic  f{x)  =  ;r*  —  ^x^  -\-6x  -{-  2  =  o.  The  first  derivative 
equation  is  /\x)  =  ^x*  —i$x'  -\-  6  =  o,  the  roots  of  which  are 
approximately  —  1.59,  —0.69,  -\-o.6g,  -\-  1.59.  Now  let  each 
of  these  values  be  substituted  for  x  in  the  given  quintic,  as  also 
the  values  —  co  ,  o,  and  +  cio ,  and  let  the  corresponding  values 
o{ /{x)  be  determined  as  follows  : 


10  THE    SOLUTION    OF    EQUATIONS.  [Chap,  I. 

;i:  =  _  00  ,      —  1.59,      —  0.69,     O,     +  0.69,      4-  1.59,      +  00  ; 
f{x)  =  -  o) ,     +  2.4,       -  0.6,  +2,     +  47,       +1.6,       +  00 . 

Since  f{x)  changes  sign  between  x^——^  and  x^^=z  —  1.59, 
one  real  root  lies  between  these  Hmits  ;  since /(;t-)  changes  sign 
between  x^  =  —1.59  and  ;i:,  =  —  0.69,  one  real  root  lies  between 
these  limits  ;  since  f{x)  changes  sign  between  x^—  —  0.69  and 
x^  =  o,  one  real  root  lies  between  these  limits;  since /(;ir)  does 
not  change  sign  between  x^  =  o  and  ;ir^  =  00  ,  a  pair  of  imagi- 
nary roots  is  indicated,  the  sum  of  which  lies  between  +  0-69 
and  00 . 

As  a  second  example  let  /(x)  =  e^  —  e^  —  4  =  0.  The  first 
derivative  equation  is  /Xx)  =  e^  —  2^"*  =  o,  which  has  two 
roots  ^*  =  i"  and  e^  =  o,  the  latter  corresponding  to  x  =  —  co , 
For  ;r  =  —  00 ,  /(x)  is  negative;  for  ^  =  ^,  /{x)  is  negative  ;  for 
X  —  -{-  <yD  ,  fix)  is  negative.  The  equation  ^—^  —  4  =  0 
has,  therefore,  no  real  roots. 

When  the  first  derivative  equation  is  not  easily  solved,  the 
second,  third,  and  following  derivatives  may  be  taken  until  an 
equation  is  found  whose  roots  may  be  obtained.  Then,  by 
working  backward,  limits  may  be  found  in  succession  for  the 
roots  oF  the  derivative  equations  until  finally  those  of  the 
primative  are  ascertained.  In  many  cases,  it  is  true,  this  proc- 
ess may  prove  lengthy  and  difficult,  and  in  some  it  may  fail 
entirely;  nevertheless  the  method  is  one  of  great  theoretical 
and  practical  value. 

Prob.  6.  Show  that  e'^  +  e~^  —4  =  0  has  two  real  roots,  one 
positive  and  one  negative. 

Prob.  7.  Show  that  ^*  +  ^+  i  =  o  has  no  real  roots;  also  that 
X*  —  X  —  I  =0  has  two  real  roots,  one  positive  and  one  negative. 


Art.  6.    Numerical  Algebraic  Equations. 

An  algebraic  equation  of  the  rf^   degree    may  be  written 
with  all  its  terms  transposed  to  the  first  member,  thus : 

x""  +  a^x""'^  +  a^x''-'^  +  ...-}-  an-\X  -\-  a^=.o\ 


Art.  6.]  numerical  algebraic  equations.  11 

and  if  all  the  coefficients  and  the  absolute  term  are  real  num- 
bers, this  is  commonly  called  a  numerical  equation.  The  first 
member  may  for  brevity  be  denoted  by  /{x)  and  the  equation 
itself  by/(;i;)  =  o. 

The  following  principles  of  the  theory  of  algebraic  equations 
with  real  coefficients,  deduced  in  text-books  on  algebra,  are 
here  recapitulated  for  convenience  of  reference : 

(i)  If  ^,  is  a  root  of  the  equation, /(;c)  is  divisible  by  ^  —  x^; 
and  conversely,  if /W  is  divisible  by  x  — :c„  then  x^  is  a  root  of  the 
equation. 

(2)  An  equation  of  the  n^^  degree  has  n  roots  and  no  more. 

(3)  If  ^, ,  ^2,  .  .  .  ^„  are  the  roots  of  the  equation,  then  the  prod- 
uct (x  —  x^)(x  —  x^)  .  .  .  (x  —  Xn)  is  equal  to/(^). 

(4)  The  sum  of  the  roots  is  equal  to  —  «,;  the  sum  of  the  prod- 
ucts of  the  roots,  taken  two  in  a  set,  is  equal  to  -\-  a^;  the  sum  of 
the  products  of  the  roots,  taken  three  in  a  set,  is  equal  to  —  ^3;  and 
so  on.  The  product  of  all  the  roots  is  equal  to  —  ^«  when  n  is 
odd,  and  to  +  «„  when  n  is  even. 

(5)  The  equation  /(x)  =  o  may  be  reduced  to  an  equation  lack- 
ing its  second  term  by  substituting  jf  —  a^/n  for  x,* 

(6)  If  an  equation  has  imaginary  roots,  they  occur  in  pairs  of 
the  form  /  ±  ^i  where  /  represents  y  —  i. 

(7)  An  equation  of  odd  degree  has  at  least  one  real  root  whose 
sign  is  opposite  to  that  of  a„. 

(8)  An  equation  of  even  degree,  having  ^«  negative,  has  at  least 
two  real  roots,  one  being  positive  and  the  other  negative. 

(9)  A  complete  equation  cannot  have  more  positive  roots  than 
variations  in  the  signs  of  its  terms,  nor  more  negative  roots  than 
permanences  in  signs.  If  all  roots  be  real,  there  are  as  many  posi- 
tive roots  as  variations,  and  as  many  negative  roots  as  permanences. f 

(10)  In  an  incomplete  equation,  if  an  even  number  of  terms, 
say  2W,  are  lacking  between  two  other  terms,  then  it  has  at  least  2Pi 

*  By  substituting  j^  -\-py-\-^  for  x,  the  quantities/ and  ^  maybe  determined 
so  as  to  remove  the  second  and  third  terms  by  means  of  a  quadratic  equation, 
the  second  and  fourth  terms  by  means  of  a  cubic  equation,  or  the  second  and 
fifth  terms  by  means  of  a  quartic  equation. 

f  The  law  deduced  by  Harriot  in  1631  and  by  Descartes  in  1639. 


12  THE    SOLUTION    OF    EQUATIONS.  [ChaP.  I. 

imaginary  roots;  if  an  odd  number  of  terms,  say  2m  +  i,  are  lacking 
between  two  other  terms,  then  it  has  at  least  either  2vi  +  2  or  2??i 
imaginary  roots,  according  as  the  two  terms  have  like  or  unlike 
signs.* 

(11)  Sturm's  theorem  gives  the  number  of  real  roots,  provided 
that  they  are  unequal,  as  also  the  number  of  real  roots  lying  be- 
tween two  assumed  values  of  x. 

(12)  If  ar  is  the  greatest  negative  coefficient,  and  if  a^  is  the 
greatest  negative  coefficient  after  x  is  changed  into  —  x^  then  all 
real  roots  lie  between  the  limits  ar-\-  \  and  —  {a^  -\-  i). 

(13)  If  ah  is  the  first  negative  and  Liy  the  greatest  negative  co- 
efficient, then  ar^~^  +  i  is  a  superior  limit  of  the  positive  roots.  If 
ak  be  the  first  negative  and  as  the  greatest  negative  coefficient  after 

X  is  changed  into  —  x,  then  a^'^  +  i  is  a  numerically  superior  limit 
of  the  negative  roots. 

(14)  Inferior  limits  of  the  positive  and  negative  roots  may  be 
found  by  placing  x  =  2;"^  and  thus  obtaining  an  equation  f{z)  =  o 
whose  roots  are  the  reciprocals  of /(x)  =  o. 

(15)  Horner's  method,  using  the  substitution  x  =  z  —  r  where  r 
is  an  approximate  value  of  x^ ,  enables  the  real  root  x^  to  be  com- 
puted to  any  required  degree  of  precision. 

The  application  of  these  principles  and  methods  will  be 
familiar  to  all  who  read  these  pages.  Horner's  method  may 
be  also  modified  so  as  to  apply  to  the  computation  of  imagi- 
nary roots  after  their  approximate  values  have  been  found. t 
The  older  method  of  Hudde  and  Rolle,  set  forth  in  Art.  5,  is 
however  one  of  frequent  convenient  application,  for  such  alge- 
braic equations  as  actually  arise  in  practice.  By  its  use, 
together  with  principles  (13)  and  (14)  above,  and  the  regula 
falsi  of  Art.  3,  the  real  roots  may  be  computed  without  any 
assumptions  whatever  regarding  their  values. 

For  example,  let  a  sphere  of  diameter  D  and  specific  gravity 

*  Established  by  Du  Gua;  see  Memoirs  Paris  Academy,  1741,  pp.  435-494. 

fSheffler,  Die  Auflosung  der  algebraischen  und  transzendenten  Gleichung- 
en,  Braunschweig,  1859;  and  Jelink,  Die  Auflosung  der  hoheren  numerischen 
Glelchungen,  Leipzig,  iSG.i^, 


Art.  7.]  TRANSCENDENTAL    EQUATIONS.  13^ 

g  float  in  water,  and  let  it  be  required  to  find  the  depth  of  im- 
mersion. The  solution  of  the  problem  gives  for  the  depth  x 
the  cubic  equation 

x'  ~\Dx'^\D'g=^o. 

As  a  particular  case  let  D  =  2  feet  and  g=  0.65  ;    then  the 

equation 

x^  —  3;r'  +  2.6  =  o 

is  to  be  solved.  The  first  derivative  equation  is  3;^'  —  6x  —  o 
whose  roots  are  o  and  2.  Substituting  these,  there  is  found 
one  negative  root,  one  positive  root  less  than  2,  and  one  posi- 
tive root  greater  than  2.  The  physical  aspect  of  the  question 
excludes  the  first  and  last  root,  and  the  second  is  to  be  computed. 
By  (13)  and  (14)  an  inferior- limit  of  this  root  is  about  0.5,  so 
that  it  lies  between  0.5  and  2.  For  x^  =  o.Sy/{x^)  =  -\-  i.g7S> 
and  for  x^  —  2,  f{x,)  =  —1.4;  then  by  the  regula  falsi  ^3=1.35. 
For  jTg  =i.35,/(-^8)  =  —  0.408,  and  combining  this  with  x,  the 
regula  falsi  gives  x^  =  1.204  ^^^t,  which,  except  in  the  last 
decimal,  is  the  correct  depth  of  immersion  of  the  sphere. 

Prob,  8.  The  diameter  of  a  water-pipe  whose  length  is  200  feet 
and  which  is  to  discharge  100  cubic  feet  per  second  under  a  head 
of  10  feet  is  given  by  the  real  root  of  the  quintic  equation 
x^  —  38JC  --  101  =  o.     Find  the  value  of  x. 

Art.  7.    Transcendental  Equations. 

Rules  (i)  to  (15)  of  the  last  article  have  no  application  to 
trigonometrical  or  exponential  equations,  but  the  general  prin- 
ciples and  methods  of  Arts.  2-5  may  be  always  used  in 
attempting  their  solution.  Transcendental  equations'  may 
have  one,  many,  or  no  real  roots,  but  those  arising  from  prob- 
lems in  physical  science  must  have  at  least  one  real  root.  Two 
examples  of  such  equations  will  be  presented. 

A  cylinder  of  specific  gravity  g  floats  in  water,  and  it  is 
required  to  find  the  immersed  arc  of  the  circumference.  If 
this  be  expressed  in  circular  measure  it  is  given  by  the  trans- 
cedental  equation 

f(^x)  =:  X  ~  sin  X  ■—  2rg  —  O. 


14  THE    SOLUTION    OF    EQUATIONS.  [Chap.  I. 

The  first  derivative  equation  is  i  —  cos  ;ir  =  c,  whose  root  is 
any  even  multiple  of  27t,  Substituting  such  multiples  in /(;tr) 
it  is  found  that  the  equation  has  but  one  real  root,  and  that 
this  lies  between  o  and  27t\  substituting  ^n,  \n,  and  n  for  x,  it 
is  further  found  that  this  root  lies  between  f tt  and  n. 

As  a  particular  case  let  g  —■  0.424,  and  for  convenience  in 
using  the  tables  let  x  be  expressed  in  degrees;  then 

fix)  —  X  —  S7°  .2958  sin  X  —  152°  .64. 
Now  proceeding  by  the  regula  falsi  (Art.  3)  let  x^  =  180°  and 
^.=  135°,  giving/(^J  =+27". 36  and/(.rj  =-58°. 16,  whence 
jTg  =166°.  For;ir3=i66°, /(;i'3)  =— o"  .469,  and  hence  166°  is  an 
approximate  value  of  the  root.  Continuing  the  process,  x  is 
found  to  be  i66°.237,  or  in  circular  measure  .;f= 2. 90 14  radians. 

As  a  second  example  let  it  be  required  to  find  the  horizon- 
tal tension  of  a  catenary  cable  whose  length  is  22  feet,  span  20 
feet,  and  weight  10  pounds  per  linear  foot,  the  ends  being  sus- 
pended from  two  points  on  the  same  level.  If  /  be  the  span,  s 
the  length  of  the  cable,  and  2  a  length  of  the  cable  whose  weight 
equals  the  horizontal  tension,  the  solution  of  the  problem  leads 

/  L        .1\ 
to  the  transcendental  equation  s=  \/*  — e  "/ ^,  or  inserting 

the  numerical  values, 

(10  I0\ 

jy^j  —  ^^         e''  —  e  ^iz  =  o 

is  the  equation  to  be  solved.     The  first  derivative  equation  is 

(  ^°         -i-°\  ,    10/12         -L°\ 

and  this  substituted  in  /(z)  shows  that  one  real  root  is  less  than 
about  20.  Assume  <sr,  =15,  then /(^,) =0.486  and /'(^,)= 0.206, 
whence  by  Newton's  rule  (Art.  4)  .s",  =  13  nearly.  Next  for 
■^■2  =  13' /(^s)  =  —  0.0298  and  /\z^)  =  0.322,  whence  ^3  =  13. i. 
Lastly  for  z^  =  13. 1  /(z^)  =0.0012  and  /'{z^)  =  0.3142,  whence 
£^  =z  13.096,  which  is  a  sufificiently  close  approximation.  The 
horizontal  tension  in  the  given  catenary  is  hence  130.96  pounds.^ 

*  Since  ^  —  e~  =2sinb6,  this  equation  may  be  written  ii6—  losinhQ, 
where  6  =  102-1,  and  the  solution  may  be  expedited  by  the  help  of  tables  of 
hyperbolic  functions.     See  Chapter  IV. 


Art.  8.]  algebraic  solutions.  15 

Prob.  9.  Show  that  the  equation  3  sin  jc  —  2^  —  5  =  o  has  but 
one  real  root,  and  compute  its  value. 

Prob.  10.  Find  the  number  of  real  roots  of  the  equation 
2x -j-  log  X  —10  000  =  o,  and  show  that  the  value  of  one  of  them  is 
^  =  4995  74- 


Art.  8.    Algebraic  Solutions. 

Algebraic  solutions  of  complete  algebraic  equations  are 
only  possible  when  the  degree  fi  is  less  than  5.  It  frequently 
happens,  moreover,  that  the  algebraic  solution  cannot  be  used 
to  determine  numerical  values  of  the  roots  as  the  formulas 
expressing  them  are  in  irreducible  imaginary  form.  Neverthe- 
less the  algebraic  solutions  of  quadratic,  cubic,  and  quartic 
equations  are  of  great  practical  value,  and  the  theory  of  the 
subject  is  of  the  highest  importance,  having  given  rise  in  fact 
to  a  large  part  of  modern  algebra. 

The  solution  of  the  quadratic  has  been  known  from  very 
early  times,  and  solutions  of  the  cubic  and  quartic  equations 
were  effected  in  the  sixteenth  century.  A  complete  investiga- 
tion of  the  fundamental  principles  of  these  solutions  was,  how- 
ever, first  given  by  Lagrange  in  1770.*  This  discussion  showed, 
if  the  general  equation  of  the  n^^  degree,  /(x)  =0,  be  deprived 
of  its  second  term,  thus  giving  the  equation /(_;V)  =  O,  that  the 
expression  for  the  root  f  is  given  by 

in  which  n  is  the  degree  of  the  given  equation,  go  is,  in  suc- 
cession, each  of  the  n^^'  roots  of  unity,  i,  e,  e^  .  .  .  e"~',  and 
s^y  s^,  .  .  .  s„_^  are  the  so-called  elements  which  in  soluble  cases 
are  determined  by  an  equation  of  the  n  —  i*^'  degree.  For 
instance,  if  ;2  =  3  the  equation  is  of  the  third  degree  or  a  cubic, 
the  three  values  of  go  are         ^ 

G?,  =  1,       G?=  —  i  +  i-/^   =  e,       GO  =   —i  —  i^^^=€\ 

*  Memoirs  of  Berlin  Academy,  1769  and  1770;  reprinted  in  CEuvres  de 
Lagrange  (Paris,  1868),  Vol.  II,  pp.  539-562.  See  also  Trait6  de  la  resolution 
des  Equations  numeriques,  Paris,  1798  and  1808. 


16  THE    SOLUTION    OF    EQUATIONS.  [ChAP.  U 

and  the  three  roots  are  expressed  by 

7i  ==  -^i  +  -^s '        J^t,  =  e5,  +  e'^,,  jj/3  =  e's,  +  e^, , 

in  which  s^^  and  s^^  are  found  to  be  the  roots  of  a  quadratic 
equation  (Art.  9). 

The  n  values  of  gd  are  the  n  roots  of  the  binomial  equation 
go"*  —  I  =0.  If  n  he  odd,  one  of  these  is  real  and  the  others 
are  imaginary ;  if  ;2  be  even,  two  are  real  and  n  —  2  are  imagi- 
nary.* Thus  the  roots  of  &?*  —  i  =  o  are  +  i  and  —  i ;  those 
of  c»'  —  I  =  o  are  given  above ;  those  of  c»*  —  i  =  o  are 
+  I,  +2,  —  I,  and  —  i  where  i  is-v/—  i.  For  the  equation 
ce?"  —  I  =  o  the  real  root  is  -|-  i,  and  the  imaginary  roots  are 
denoted  by  e,  e^  e^  e*;  to  find  these  let  Gt>^  —  i  =0  be  divided 
by  QD—  I,  giving 

Od"  -\-  00^  -\-  QD^  -\-  OD  A-  \  =0, 

which  being  a  reciprocal  equation  can  be  reduced  to  a  quad- 
ratic, and  the  solution  of  this  furnishes  the  four  values, 

e    =-^(1-^/5  +    |/_io-2V5),       e*  =  _^(i+  V5  +   V-   10+  2V5)r 

e*  =  -i(i-V5-   |/_io-2V5),     e'  =  -  i(i  +  'V's  —  '^^-  10  +  2  V5), 

where  it  will  be  seen  that  e.e*  =  I  and  e^e''  =  I,  as  should  be 
the  case,  since  e^  =  i. 

In  order  to  solve  a  quadratic  equation  by  this  general 
method  let  it  be  of  the  form 

x^  +  2<3:;r  -[-  ^  =  o, 

and  let  x  be  replaced  hy  y  —  a,  thus  reducing  it  to 

f  _  {a'  -b)  =  o. 

Now  the  two  roots  of  this  are  y^^  -\-  s^  and  j,  =  —  j^,  whence 
the  product  of  {y  —  s^  and  {y  -f-  s^  is 

/  -s'  =  o. 

Thus  the  value  of  /  is  given  by  an  equation  of  the  first  degree, 

*  The  values  of  go  are,  in  short,  those  of  the  n  "  vectors  "  drawn  from  the 
center  which  divide  a  circle  of  radius  unity  into  n  equal  parts,  the  first  vector 
GJi  =  I  being  measured  on  the  axis  of  real  quantities.     See  Chapter  X. 


Art.  9.]  THE  cubic  equation.  17 

s^=.a^  —  b\  and  since  x^  —^-\-fy  the  roots  of  the  given 
equation  are 

x^  =  —  a-\-  -yj  a'  —  b,  x^  =  —  a  —  ^a'  —  bj 

which  is  the  algebraic  solution  of  the  quadratic. 

The  equation  of  the  n  —  i^^  degree  upon  which  the  solution 
of  the  equation  of  the  n^^  degree  depends  is  called  a  resolvent. 
If  such  a  resolvent  exists,  the  given  equation  is  algebraically 
solvable ;  but,  as  before  remarked,  this  is  only  the  case  for 
quadratic,  cubic,  and  quartic  equations, 

Prob.  II.  Show  that  the  six  6^^  roots  of  unity  are  +  i, 
+4(1+  ^~3),  -«i-  ^^3),  -I,   -4(i+  ^^3),  -4(i-  ^=3). 

Art.  9.    The  Cubic  Equation. 

All  methods  for  the  solution  of  the  cubic  equation  lead  to 
the  result  commonly  known  as  Cardan's  formula.*     Let  the 

cubic  be 

x' -{- 7,ax' +  ^bx -{- 2c  =  o,  (i) 

and  let  the  second  term  be  removed  by  substituting  7  —  a  for 

X,  giving  the  form, 

y  +  SBy+2C=o,  (I') 

in  which  the  values  of  B  and  C  are 

B=-a'  +  b,         C=a'-iab  +  c.  (2) 

Now  by  the  Lagrangian  method  of  Art.  8  the  values  of  j/  are 

I^=  s,  +  s,,        y.,  =  es,  +  e\,         y,  =  e's,  +  es,, 

in  which  e  and  e'  are  the  imaginary  cube  roots  of  unity. 
Forming  the  products  of  the  roots,  and  remembering  that 
6^  =  I  and  e^  +  e"  +  I  =0,  there  are  found 

J,7,  +  J,7.  +7J^3  =  -  3S.S,  =  +  sB, 
J.J^,J3  =  ^,' +  V  =  -2^. 
For  the  determination  of  j,  and  s^  there  are  hence  two  equa- 
tions from  which  results  the  quadratic  resolvent 
/  +  2Cs'  —  B'  =  o,  and  thus 

s,={-C+VB^+T'y,     s,  =  {-C-VBT~C'y.     (3) 

*  Deduced  by  Ferreo  in  1515,  and  first  published  by  Cardan  in  1545. 


18  THE    SOLUTION    OF    EQUATIONS.  [ChAP.  I. 

One  of  the  roots  of  the  cubic  in  j/  therefore  is 

and  this  is  the  well-known  formula  of  Cardan. 

The  algebraic  solution  of  the  cubic  equation  (i)  hence  con- 
sists in  finding  B  and  C  by  (2)  in  terms  of  the  given  coefficients, 
and  then  by  (3)  the  elements  s^  and  s^  are  determined.    Finally, 

:^,  =  -  a+  {s,+s,), 

^,  =  -a- i{s,  +  s,)  +  i^~:r~^(^s,  - s,),  (4) 

^,=  -a-  i{s,  4-  s,)  -  ^^~:^^{s,  -  s,\ 

which  are  the  algebraic  expressions  of  the  three  roots. 

When  B^ -\- C^  is  negative  the  numerical  solution  of  the 
cubic  is  not  possible  by  these  formulas,  as  then  both  s^  and  s, 
are  in  irreducible  imaginary  form.  This,  as  is  well  known,  is 
the  case  of  three  real  roots,  j,  -|-  s^  being  a  real,  while  s^  —  s^  is 
a  pure  imaginary.*  When  B""  +  C^  is  o  the  elements  j,  and  s^ 
are  equal,  and  there  are  two  equal  roots,  x^  =  x^  ^  —  a  -{-  C^^ 
while  the  other  root  is  ;r,  =  —  a  —  2C^. 

When  B^ -\-  C  is  positive  the  equation  has  one  real  and 
two  imaginary  roots,  and  formulas  (2),  (3),  and  (4)  furnish  the 
numerical  values  of  the  roots  of  (i).  For  example,  take  the 
cubic 

x' —  4.sx^ -\- 12X  —  s  =  o, 

whence  by  comparison  with  (i)  are  found  ^  =  —  1.5,  <^  =:  -|-  4, 

c  —  —2.5.     Then  from  (2)  are  computed  B  =  1.75,  6^=-|-3.i25. 

These  values  inserted  in  (3)  give  ^j  =  -|- 0.9142,  s^  =  —  1.9142  ; 

thus  J,  +  5,  =  —  i.o  and  ^,—  ^^  ==  +  2.8284.     Finally,  from  (4) 
X,  =  1.5  -  1.0  =  +0.5, 

X,  =  1.5  +0.5  +  1.4142  4/~=^3  =  2  +  2.4495/, 
X,  =  1.5  +  0.5  -  1.4142  V^^  =  2-  2.4495/, 

which  are  the  three  roots  of  the  given  cubic. 

*  The  numerical  solution  of  this  case  is  possible  whenever  the  angle  whose 
icosine  is  —  C/  \/—  B'^  r-^n  be  sfeometrically  trisected. 


Art.  10.]  THE    QUARTIC    EQUATION.  19 

Prob.  12.  Compute  the  roots  of  x'  —  2x—^=  o.  Also  the  roots 
of  x^  -\-  o.Gx"^  —  $.t6x  4-  4.32  =  o. 

Prob.  13.  A  cone  has  its  altitude  6  inches  and  the  diameter  of 
its  base  5  inches.  It  is  placed  with  vertex  downwards  and  one  fifth 
of  its  volume  is  filled  with  water.  If  a  sphere  4  inches  in  diameter 
be  then  put  into  the  cone,  what  part  of  its  radius  is  immersed  in  the 
water  ?     (Ans.  0.5459  inches). 

Art.  10.    The  Quartic  Equation. 

The  quartic  equation  was  first  solved  in  1545  by  Ferrari, 
who  separated  it  into  the  difference  of  two  squares.  Lagrange 
in  1637  resolved  it  into  the  product  of  two  quadratic  factors. 
Tschirnhausen  in  1683  removed  the  second  and  fourth  terms. 
Euler  in  1732  and  Lagrange  in  1767  effected  solutions  by 
assuming  the  form  of  the  roots.  All  these  methods  lead  to 
cubic  resolvents,  the  roots  of  which  are  first  to  be  found  in 
order  to  determine  those  of  the  quartic. 

The  methods  of  Euler  and  Lagrange,  which  are  closely 
similar,  first  reduce  the  quartic  to  one  lacking  the  second  term, 

/  +  6Bf  +  4(5/  +  Z>  =  o ; 

and  the  general  form  of  the  roots  being  taken  as 

J,  =  +  ^^  +  ^^.  +  ^^3.         y^  =  -^^^■.  +  ^^.-  V7., 
y.  =  +  ^^^-VJ,-  V7,,         y^  =  -VJ^-  V7,  +  V's], 

the  values  j,,  j„  j,,  are  shown  to  be  the  roots  of  the  resolvent, 

/  +  2>Bs'  +  i(9^'  -  D)s  -  \r  =  o. 

Thus  the  roots  of  the  quartic  are  algebraically  expressed  in 
terms  of  the  coefficients  of  the  quartic,  since  the  resolvent  is 
solvable  by  the  process  of  Art.  9. 

Whatever  method  of  solution  be  followed,  the  following 
final  formulas,  deduced  by  the  author  in  1892,  will  result.* 
Let  the  complete  quartic  equation  be  written  in  the  form 

X*  +  4ax'  +  6dx'  ■\- a,cx  ^  d  =  o.  (i) ' 

*  See  American  Journal  Mathematics,  1892,  Vol.  XIV,  pp.  237-245. 


20  THE    SOLUTION    OF    EQUATIONS.  [ChAP.  L 

First,  let  g,  h,  and  k  be  determined  from 

g^a'-b,     h^b'  +  e  -2abc-\-dg,     k=iac-d'-id.   (2) 

Secondly,  let  /  be  obtained  by 

/  =  i{k  +  V¥~+Ff  +  i{/i  -  VF+I'Y  (3) 

Thirdly,  let  u,  v,  and  w  be  found  from 

u=g+l,     v  =  2g-l,     w  =  4u'  +  3k-  I2gl,  (4) 

Then  the  four  roots  of  the  quartic  equation  are 


x,=  —a+Vu-\-^v  +  V 


Wy 


x^=  —  a  -{-  Vu  —  "^V  ■r{-  VWy 


x^=  —  a  —  Vu  -\-  yv  —  Vzu, 


y         (5) 


x^=  —  a  —  Vu  —  "yv  —  Vw,  ^ 
in  which  the  signs  are  to  be  used  as  written  provided   that 
2a^  —  sab  -|-^  is  a  negative  number;  but  if  this  is  positive  all 
radicals  except  Vw  are  to  be  reversed  in  sign. 

These  formulas  not  only  serve  for  the  complete  theoretic 
discussion  of  the  quartic  (i),  but  they  enable  numerical  solu- 
tions to  be  made  whenever  (3)  can  be  computed,  that  is,  when- 
ever /i''  -\-k^  is  positive.  For  this  case  the  quartic  has  two  real 
and  two  imaginary  roots.  If  there  be  either  four  real  roots  or 
four  imaginary  roots  h^  +  k""  is  negative,  and  the  irreducible 
case  arises  where  convenient  numerical  values  cannot  be  ob- 
tained, although  they  are  correctly  represented  by  the  formulas. 

As  an  example  let  a  given  rectangle  have  the  sides/  and  q, 
and  let  it  be  required  to  find  the  length  of  an  inscribed  rec- 
tangle whose  width  is  m.  If  x  be  this  length,  this  is  a  root  of 
the  quartic  equation 

x'  -  (D''  +  ^'  +  2ne)x'  +  4Pqmx  -  (/  +  ^'  -  ne)ni'  =  o, 

and  thus  the  problem  is   numerically  solvable   by  the  above 

formulas  if  two  roots  are  real  and  two  imaginary.     As  a  special 

•  case  let /  =  4  feet,  g=  s  ^ eet,  and  7n  =  i  foot ;  then 

X*  —  2yx''  +  48;ir  _  24  =  o. 


Art.  11.]  QUINTIC    EQUATIONS.  21 

By  comparison  with  (i)  are  found  ^  =  o,  <5  =  —  4^,  ^  =  +  12, 
and  d=—24.  Then  from  (2),  £  =  +4h  ^  =  —H-y  and 
k=  -{-  "Y-.  Thus  k*  +  k^  is  positive,  and  from  (3)  the  value  of  / 
15—3.6067.  From  (4)  are  now  found,  z^=: +0.8933,  z;=  12.6067, 
and  w=  -{-  161.20.  Then,  since  c  is  positive,  the  values  of  the 
four  roots  are,  by  (5), 


x,=  —  0.945  —  4/12.607  +  12.697  =  —  5.975  feet, 
x^=  —  0.945  +  Vi 2.607+  12.697  =  +  4.085  feet, 
x,=  -  0.945  +  ^12.607-  12.697  =  +  0.945  +  0.30^; 
x^=  —  o  945  —  V 1 2.607  —  12.697  =  +  0.945  —  0.302, 
the  second  of  which  is 'evidently  the  required  length.     Each  of 
these  roots  closely  satisfies  the  given  equation,  the  slight  dis- 
crepancy in  each  case  being  due  to  the  rounding  off  at  the  third 
decimal.* 

Prob.  14.  Compute  the  roots  of  the  equation  x*  -{-  'jx  -\-  6  =  o, 
(Ans.  —  1.388,  —  1.000,  1. 194  ±  1. 701/.) 


Art.  11.    QuiNTic  Equations. 

The  complete  equation  of  the  fifth  degree  is  not  algebraic- 
ally solvable,  nor  is  it  reducible  to  a  solvable  form.  Let  the 
equation  be 

x"  +  $ax*  +  Sdx^  +  5^;tr*  +  ^dx  +  2^  =r  O, 

and  by  substituting  j  —  ^  for  ;tr  let  it  be  reduced  to 
The  five  roots  of  this  are,  according  to  Art.  8, 

J^a  =  e-^i  +  e\  +  6^3  +  eV„ 

.y,  =  e%  +  €\  +  e^3  +  €%, 

y.  =  e's,  +  e^,  +  €\  +  eV,, 

^6  =  eV,  +  e%  +  eX  +  e^„ 
in  which  e,  €*,  e".  e*  are  the  imaginary  fifth  roots  of  unity.     Now 
if  the  several  products  of  these  roots  be  taken  there  will  be 

*  This  example  is  known  by  civil  engineers  as  the  problem  of  finding  the 
length  of  a  strut  in  a  panel  of  the  Howe  truss. 


22  THE    SOLUTION    OF    EQUATIONS.  [ChAP.  I. 

found,  by  (4)  of  Art.  6,  four  equations  connecting  the  four  ele- 
ments ^1,  s^,  ^3,  and  s^,  namely, 

-  B  =  s,s,  +  s^s,, 

—  D  =  s,\  +  s,'s,  +  ^3'^,  +  ^,^3  —  s's'  —  s,\  +  s,s,s,s^, 

but  the  solution  of  these  leads  to  an  equation  of  the  I20th 
degree  for  s,  or  of  the  24th  degree  for  s^.  However,  by  taking 
s^s^  —  V3  or  ^1"  +  -^n'  +  -^3'  +  -^4^  ^s  the  unknown  quantity,  a 
resolvent  of  the  6th  degree  is  obtained,  and  all  efforts  to  find 
a  resolvent  of  the  fourth  degree  have  proved  unavaiHng. 

Another  line  of  attack  upon  the  quintic  is  in  attempting  to 
remove  all  the  terms  intermediate  between  the  first  and  the 
last.  By  substituting  y  -\- py  -\-q  for  x,  the  values  of />  and  q 
may  be  determined  so  as  to  remove  the  second  and  third  terms 
by  a  quadratic  equation,  or  the  second  and  third  by  a  cubic 
equation,  or  the  second  and  fourth  by  a  quartic  equation,  as 
was  first  shown  by  Tschirnhausen  in  1683.  By  substituting 
y  +/'y  ~\~  qy  -\-'^  for  X,  three  terms  may  be  removed,  as  was 
shown  by  Bring  in  1786.  By  substituting  y-f-/y-(- ^y-|-0^+^ 
for  X  it  was  thought  by  Jerrard  in  1833  that  four  terms  might 
be  removed,  but  Hamilton  showed  later  that  this  leads  to 
equations  of  a  degree  higher  than  the  fourth. 

In  1826  Abel  gave  a  demonstration  that  the  algebraic  solu- 
tion of  the  general  quintic  is  impossible,  and  later  Galois 
published  a  more  extended  investigation  leading  to  the  same 
conclusion.*  Although  these  discussions  are  complex,  and  not 
devoid  of  points  of  doubt,t  they  have  been  generally  accepted 
as  conclusive.  Moreover,  the  fact  that  the  quintic  is  still  un- 
solved in  spite  of  the  enormous  amount  of  work  done  upon  it 
during  the  past  two  centuries,  is  strong  evidence  that  the  prob- 
lem is  an  impossible  one. 

*See  Jordan's  Trait6  des  substitutions  et  des  Equations  alg6briques,  1870. 
f  See  Kronecker,  Verhandlungen  der  Berliner  Akademie,  1853,  p.  38;  also- 
Cockle,  Philosophical  Magazine,  1854,  Vol.  VII,  p.  134. 


Art.  11.]  QUINTIC    EQUATIONS.  23 

There  are,  however,  numerous  special  forms  of  the  quintic 
whose  algebraic  solution  is  possible.  The  oldest  of  these  is  the 
quintic  of  De  Moivre,  (^ 

which  is  solved  at  once  by  making  s^  =  s^  =  o  in  the  element 
equations  ;  then  —  B  —  s^s^  and  —  2E  =  .y/  +  •^4'»  ^^om  which 
s^  and  s^  are  found,  and  y^  =  s^-\-s^,  or 

y,  ={-  E  +  VB^+~Wy  -\-{-E-  VB^+E^y, 
while    the    other    roots    are   y^  =  es,  -)-  ^'^^ ,  j,  =  e'.y,  +  €^s^ , 
y^=  eV,  +  e\»   ^"d  /ft  ^  ^*^i  +  ^-^4  •     If  ^'  +  ^'  be  negative, 
this  quintic  has  five  real  roots;  if  positive,  there  are  one  real 
and  four  imaginary  roots. 

When  any  relation,  other  than  those  expressed  by  the  four 
element  equations,  exists  between  s^,  s^,  s^,  s^y  the  quintic  is 
solvable  algebraically.  As  an  infinite  number  of  such  relations 
may  be  stated,  it  follows  that  there  are  an  infinite  number  of 
solvable  quintics.  In  each  case  of  this  kind,  however,  the  co- 
efficients of  the  quintic  are  also  related  to  each  other  by  a 
certain  equation  of  condition. 

The  complete  solution  of  the  quintic  in  terms  of  one  of  the 
roots  of  its  resolvent  sextic  was  made  by  McClintock  in  1884.* 
By  this  method  s^\  s^\  s^^j  and  .y/  are  expressed  as  the  roots  of 
a  quartic  in  terms  of  a  quantity  t  which  is  the  root  of  a  sextic 
whose  coefficients  are  rational  functions  of  those  of  the  given 
quintic.  Although  this  has  great  theoretic  interest,  it  is,  of 
course,  of  little  practical  value  for  the  determination  of  numer- 
ical values  of  the  roots. 

By  means  of  elliptic  functions  the  complete  quintic  can, 
however,  be  solved,  as  was  first  shown  by  Hermite  in  1858. 
For  this  purpose  the  quintic  is  reduced  by  Jerrard's  transfor- 
mation to  the  form  x"  -{-  ^dx-\-2e  =  o,  and  to  this  form  can 
also  be  reduced  the  eUiptic  modular  equation  of  the  sixth 
degree.     Other  solutions  by  elliptic  functions  were  made  by 

*  American  Journal  of  Mathematics,  1886,  Vol.  VIII,  pp.  49-83. 


24  THE    SOLUTION    OF    EQUATIONS.  [ChAP.  I. 

Kronecker  in  1861  and  by  Klein  in  1884.'^  These  methods, 
though  feasible  by  the  help  of  tables,  have  not  yet  been  sys- 
tematized so  as  to  be  of  practical  advantage  in  the  numerical 
computation  of  roots. 

Prob.  15.  If  the  relation  s^s^  =s^s,  exists,  between  the  elements 
show  that  s,'  +  s^'  +  ^3'  +  j/  ==  -  2jS. 

Prob.  16.  Compute  the  roots  of  y^  +  loy  +  2oy  +  6  =  0,  and 
also  those  of  ^^  —  10^  +  2qy  +  6  =  o. 

Art.  12.    Trigonometric  Solutions. 

When  a  cubic  equation  has  three  real  roots  the  most  con- 
venient practical  method  of  solution  is  by  the  use  of  a  table  of 
sines  and  cosines.  If  the  cubic  be  stated  in  the  form  (i)  of 
Art.  9,  let  the  second  term  be  removed,  giving 

Now  suppose  y=^2r  sin   d,  then  this  equation  becomes 

B  C 

8  sin' ^+6-3  sin  6^+2-^  =  0, 

and  by  comparison  with  the  known  trigonometric  formula 

8  sin'  ^  —  6  sin  ^  -j-  2  sin  3^  =  O, 

there  are  found  for  r  and  sin  3^  the  values 

r  =  V—'W,  sin  36>  =  C/  V^^T^, 

in  which  B  is  always  negative  for  the  case  of  three  real  roots 
(Art.  9).  Now  sin  3^  being  computed,  36^  is  found  from  a  table 
of  sines,  and  then  6  is  known.     Thus, 

y^  =  2r  sin  ^,  jj/,  =  2r  sin  (120°  +  (9),  j|/,  =  2r  sin  (240**  +  ff), 
are  the  real  roots  of  the  cubic  in  jj/.f 

*  For  an  outline  of  these  transcendental  methods,  see  Hagen's  Synopsis  der 
hoheren  Mathematik,  Vol.  I,  pp.  339-344. 

f  When  B^  is  negative  and  numerically  less  than  C'^,  as  also  when  B^  is 
positive,  this  solution  fails,  as  then  one  root  is  real  and  two  are  im.aginary.  In 
this  case,  however,  a  similar  method  of  solution  by  means  of  hyperbolic  sines 
is  possible.  See  Grunert's  Archiv  fiir  Mathematik  und  Physik,  Vol.  xxxviii, 
pp.  48-76. 


Art.  12.]  trigonometric  solutions.  25 

For  example,  the  depth  of  flotation  of  a  sphere  whose  diam- 
eter is  2  feet  and  specific  gravity  0.65,  is  given  by  the  cubic 
equation  x^  —  3;^'  +  2.6  =  o  (Art.  6).  Placing  x  =  y  -\-  i  this 
reduces  to  y  —  3 J  +  0.6  =  o,  for  which  B  =  —  i  and  C  =r-fo.3. 
Thus  r=  I  and  sin  ^0  =:-|-o.3.  Next  from  a  table  of  sines, 
319  =  17°  2fy  and  accordingly  0  =  S°  49'.     Then 

^,  =  2  sin      5°  49'  —  +0.2027, 

J,  =  2  sin  125°  49'  =  +  1.6218, 

j^,  =  2  sin  245°  49'  =  —  1.8245. 

Adding  I  to  each  of  these,  the  values  of  x  are 

X,  =-{- 1.203  feet,    ^a  =  +  2.622  feet,    x,  =—0.825  feet ; 

and  evidently,  from  the  physical  aspect  of  the  question,  the 
first  of  these  is  the  required  depth.  It  may  be  noted  that  the 
number  0.3  is  also  the  sine  of  162°  11',  but  by  using  this  the 
three  roots  have  the  same  values  in  a  different  order. 

When  the  quartic  equation  has  four  real  roots  its  cubic  re- 
solvent has  also  three  real  roots.  In  this  case  the  formulas  of 
Art.  10  will  furnish  the  solution  if  the  three  values  of  /  be  ob- 
tained from  (3)  by  the  help  of  a  table  of  sines.  The  quartic 
being  given,  ^,  ^,  and  k  are  found  as  before,  and  the  value  of 
^  will  always  be  negative  for  four  real  roots.     Then 


rz=z  V—k,  sin  30=  —  h/r\ 

and  3^  is  taken  from  a  table  ;  thus  B  is.known,  and  the  three 
values  of  /  are 

/,  =  r  sin  e,  h  —  r  sin(i20°  +  e\  /,  =  ^  sin  (240°  +  6). 
Next  the  three  values  of  «,  of  v,  and  of  w  are  computed,  and 
those  selected  which  give  u^  w,  and  v  —  Vw  all  positive  quanti- 
ties.    Then  (5)  gives  the  required  roots  of  the  quartic. 

As  an  example,  take  the  case  of  the  inscribed  rectangle  in 
Art.  10,  and  let/  =  4  feet,  q  =  3  feet,  m  =  1^13  feet;  then  the 
quartic  equation  is 

X*  —  ^  IX*  +  48  VYs  X  —  156  =  o. 


26  THE    SOLUTION    OF    EQUATIONS.  [Chap.  L 

Here  a  =  o.  5  =  —  S^,  c  =  -{-12  ViS,  and  d  =  —  156.  Next 
^  =+  SJ,  /i=  —  -S-f-^,  and  k  =:—  ^'  The  trigonometric  work 
now  begins;  the  value  of  r  is  found  to  be  +  4J,  and  that  of 
sin  36^  to  be  +  0.7476;  hence  from  the  table  36*  =  48°  23',  and 
6=  i6°07'4o'^  The  three  values  of  /  are  then  computed 
by  logarithmic  tables,  and  found  to  be, 

/,  =  +  1.250,  /,  =  +  3.1187,  /,  =  -4.3687. 

Next  the  values  of  ?/,  v,  and  w  are  obtained,  and  it  is  seen  that 
only  those  corresponding  to  /,  will  render  all  quantities  under 
the  radicals  positive  ;  these  quantities  are  u  =  9.75,  v  =  iS'7Sr 
and  w  =  192.0.     Then  the  four  roots  of  the  quartic  are 

^i  =  -8.564,  ^Q  =+  2.319'  -^3=+  1.746,  ^4  =  +  4499  ^eet, 
of  which  only  the  second  and  third  belong  to  inscribed  rec- 
tangles, while  the  first  and  fourth  belong  to  rectangles  whose 
corners  are  on  the  sides  of  the  given  rectangle  produced. 

Trigonometric  solutions  of  the  quintic  equation  are  not 
possible  except  for  the  binomial  x''  ±  a,  and  the  quintic  of 
De  Moivre.  The  general  trigonometric  expression  for  the  root 
iof  a  quintic  lacking  its  second  term  isjj/=2r,  cos  ^,+2^2  cos  ^, ,. 
and  to  render  a  solution  possible,  r,  and  r^ ,  as  well  as  cos  6^ 
and  cos  6^3,  must  be  found;  but  these  in  general  are  roots  of 
equations  of  the  sixth  or  twelfth  degree :  in  fact  r^'  is  the  same 
as  the  function  s^s^  of  Art.  11,  and  r/  is  the  same  as  s^s^. 
Here  cos^,  and  cos  6^3  may  be  either  circular  or  hyperbolic 
cosines,  depending  upon  the  signs  and  values  of  the  coefficients 
of  the  quintic. 

Trigonometric  solutions  are  possible  for  any  binomial  equa- 
tion, and  also  for  any  equation  which  expresses  the  division  of 
an  angle  into  equal  parts.  Thus  the  roots  of  ^tr"  +  i  =0  are 
cosm  30°  ±  t  sin  ;;/  30°,  in  which  m  has  the  values  i,  2,  and  3. 
The  roots  of  x^'—^x^'-^-^x  —  2  cos  5  6^  =  o  are  2  cos  {in  'j2°-\-d) 
where  in  has  the  values  o,  i,  2,  3,  and  4. 

Prob.  17.  Compute  by  a  trigonometric  solution  the  four  roots  of 
the  quartic^*  +  ^x^  —  24^^  —  76:^  —  29  =  o.  (Ans.  —6.734,  -1.550^ 
+  0.262,  +  4.022). 


Art.  13.]  REAL    ROOTS    BY    SERIES.  27 

Prob.  1 8.  Give  a  trigonometric  solution  of  the  quvntic  equation 
x^  —  ^bx^  -\-  ^b'^x  —  2^  =  o  for  the  case  of  five  real  roots.  Compute 
the  roots  when  b  =i  and  e  =  0.752798.  (Ans.  —1.7940,  —  1.3952, 
0.2864,  0.9317,  1. 9710.) 

Art.  13.    Real  Roots  by  Series. 

The  value  of  x  in  any  algebraic  equation  may  be  expressed 
as  an  infinite  series.  Let  the  equation  be  of  any  degree,  and 
by  dividing  by  the  coefificient  of  the  term  containing  the  first 
power  of  X  let  it  be  placed  in  the  form 

a  =  x+dx'  +  cx'+  dx'  +  ex' -f /r«-(- .  .  . 
Now  let  it  be  assumed  that  x  can  be  expressed  by  the  series 

X  ^  a-\-  ma^  -\-  na^  +/^*  +  q<^^  +  •  •  • 
By  inserting  this  value  of  x  in  the  equation  and  equating  the 
coefficients  of  like  powers  of  «,  the  values  of  m,  n,  etc.,  are 
found,  and  then 

x=a  —  ba'  +  {2b''—c)a^—(<^b^—Sbc-\-d)a'  +  {i^b*—2ib\^-6bd^-2,c''—e)a!' 

-{\2b'  -  84^V  +  28W  +  2Uc''  -  ^be  -  1cd-\-f)a'  + .  .  . , 
is  an  expression  of  one  of  the  roots  of  the  equation.     In  order 
that  this  series  may  converge  rapidly  it  is  necessary  that   a 
should  be  a  small  fraction.* 

To  apply  this  to  a  cubic  equation  the  coefficients  d,  e,f,  etc., 
are  made  equal  to  o.  For  example,  let  x^  —  3;tr  +  0.6  =  o  ; 
this  reduced  to  the  given  form  is  0.2  =  ;i;  —  Jjtr',  hence  a  =  0.2, 
b  =  o,  c  =  —  -J,  and  then 

x  =  o.2-\-i,  0.2'  +  i .  0.2'  +  etc.  =  +  0.20277, 

which  is  the  value  of  one  of  the  roots  correct  to  the  fourth 
decimal  place.  This  equation  has  three  real  roots,  but  the 
series  gives  only  one  of  them  ;  the  others  can,  however,  be 
found  if  their  approximate  values  are  known.  Thus,  one  root 
is  about  +1.6,  and  by  placing  x=j/-\'i.6  there  results  an 
equation  in  y  whose  root  by  the  series  is  found  to  be  +  0.0218, 
and  hence  -\-  1.62 18  is  another  root  of  x^  —  ^x  -\-  0.6  =  o. 

*This  method  is  given  by  J.  B.  Mott  in  The  Analyst,  1882,  Vol.  IX,  p.  104.. 


;28  THE    SOLUTION    OF    EQUATIONS.  [Chap.  I. 

Cardan's  expression  for  the  root  of  a  cubic  equation  can  be 
expressed  as  a  series  by  developing  each  of  the  cube  roots  by 
the  binomial  formula  and  adding  the  results.  Let  the  equa- 
tion be  y  +  iBy  -\-  2C  =  o,  whose  root  is,  by  Art.  9, 

j;=:(-  c+VB^+c-y+i-  C-  s/wvcf. 

then  this  development  gives  the  series, 

,       ^^y         2  2.5.8,       2. 5. 8.  II . 14    3  \ 

y  —  2(—  C^\  \  —  -r r" ^  r'  —  .  .  .  j, 

-^         ^      ^^\         2  2.3.4  2.3.4.5.6  /' 

in  which  r  represents  the  quantity  {B^  -\-  C^)/iC\  li  r  —  o 
the  equation  has  two  equal  roots  and  the  third  root  is  2{~C )^. 
If  r  is  numerically  greater  than  unity  the  series  is  divergent, 
and  the  solution  fails.  If  r  is  numerically  less  than  unity  and 
sufficiently  small  to  make  a  quick  convergence,  the  series  will 
serve  for  the  computation  of  one  real  root.  For  example,  take 
the  equation  x^  —  6x-\-6=o,  where  B=  —  2  and  (7*=:  3; 
hence  r  =  1/8 1,  and  one  root  is 
J  =  —  2.8845(1  —  0.01235  —  0.00051  —  0.00032—)  =  —  2.846, 

which  is  correct  to  the  third  decimal.     In  comparatively  few 
cases,  however,  is  this  series  of  value  for  the  solution  of  cubics. 

Many  other  series  for  the  expression  of  the  roots  of  equa- 
tions, particularly  for  trinomial  equations,  have  been  devised. 
One  of  the  oldest  is  that  given  by  Lambert  in  1758,  whereby 
the  root  of  x""  -{-  ax  —  d  =  o  is  developed  in  terms  of  the 
ascending  powers  of  d/a.  Other  solutions  were  published  by 
Euler  and  Lagrange.  These  series  usually  give  but  one  root, 
and  this  only  when  the  values  of  the  coefficients  are  such  as  to 
render  convergence  rapid. 

Prob.  19.  Consult  Euler's  Anleitung  zur  Algebra  (St.  Petersburg, 
1771),  pp.  143-150,  and  apply  his  method  of  series  to  the  solution  of 
a  quartic  equation. 

Art.  14.     Computation  of  all  Roots. 

A  comprehensive  and  valuable  method  for  the  solution  of 
•equations  by  series  was  developed  by  McClintock,  in  1894,  by 


Art.  14.]  COMPUTATION    OF    ALL    ROOTS  29*' 

means  of  his  Calculus  of  Enlargement.*  By  this  method  all 
the  roots,  whether  real  or  imaginary,  may  be  computed  from  a 
single  series.  The  following  is  a  statement  of  the  method  as 
applied  to  trinomial  equations : 

Let  x"*  =  nAx^'^  -\-  B^   ho.  the   given   trinomial   equation. 
Substitute  x  =  By  and  thus  reduce  the  equation  to  the  form 
^«  =  nay'"^-\-\  where  a  =  A/B^.     Then  if  B*"  is  positive,  the 
roots  are  given  by  the  series 

y  —  Qo-\-  GD'-^  a  +  c»'-^*(i  —2k-\-  n)a^l2 1 
+(»^-3^(i-3^+«)(i-3>^+2«>y3! 
-{-Go'-^{i—4k-]-n){i—4k-{-2n){i—4k+^n)a*/4l  + .  . . ^ 

in  which  qi  represents  in  succession  each  of  the  roots  of  unity. 
If,  however,  B""  is  negative,  the  given  equation  reduces  to 
y"*  z=z  nay*"'^  —  I ,  and  the  same  series  gives  the  roots  if  gd  be 
taken  in  succession  as  each  of  the  roots  of  —  I. 

In  order  that  this  series  may  be  convergent  the  value  of  a" 
must  be  numerically  less  than  k~\n  —  k)^'*" ;  thus  for  the  quar- 
tic  y*  =  4ax  +  I,  where  n  =  4  and  ^  =  3,  the  value  of  a  must 
be  less  than  27~*. 

To  apply  this  method  to  the  cubic  equation  x^=^Ax±B\ 
place  «  =  3  and  ^  =  2,  and  put  y  =  Bx.  It  then  becomes 
y  =  ^ay  ±  I  where  a  =  A/B\  and  the  series  is 

y  =  Gj-\-  Go^a  —  -Jg?^'  +  Jfi?'^*  +  •  •  •  > 

in  which  the  values  to  be  taken  for  go  are  the  cube  roots  of  r 
or  —  I,  as  the  case  may  be.  For  example,  let  x^—  2x  —  ^  =0. 
Placing  jj/=  5^^,  this  reduces  toy =0.684 jj/+i.  Here  ^=0.228, 
and  as  this  is  less  than  4-*  the  series  is  convergent.  Making; 
<w  =  I,  the  first  root  is 

y  =  I  +0.2280  —  0.0039  +  0.0009  =  1.2250. 

*See  Bulletin  of  American  Mathematical  Society,  1894,  Vol.  I,  p.  3;  alsa 
American  Journal  of  Mathematics,  1895,  V9I.  XVII,  pp.  89-110. 


30  THE    SOLUTION    OF    EQUATIONS.  [CkAP,  1. 

Next  making  go  =  —  i  +  i  t^^,  a?"  is  —  J  —J  V—  3, 
and  the  corresponding  root  is  found  to  be 

j/=  -  0.6125  +  0.3836  V^, 


Again,  making  &?  =  —  i  —  4  V—  3  the  third  root  is  found  to 
be  the  conjugate  imaginary  of  the  second.    Lastly,  multiplying 

each  value  of  jj/  by  5*,  

X  =  2.095,  :i;=  -  1.047  ±  1-136  V—  I, 

which  are  very  nearly  the  roots  oi  x^  —  2x  —  ^  =  o. 

In  a  similar  manner  the  cubic  x^  -\-  2x  -\-  $  =0  reduces  to 
y  =  —  0.684J  —I,  for  which  the  series  is  convergent.     Here 
the  three  values  of  go  are,   in  succession,  —  i,  ^  +i  V—  3, 
_  ^  _L  J.  |/_  3,    and    the   three    roots   are  /  =  —  0.777   and 
J/ =  0.388  ±  1.137^*- 

When  all  the  roots  are  real,  the  method  as  above  stated 
fails  because  the  series  is  divergent.  The  given  equation  can, 
however,  be  transformed  so  as  to  obtain  n  —  k  roots  by  one 
application  of  the  general  series  and  k  roots  by  another.  As 
an  example,  let  x"  —  2\ix  +  330  =  o.  For  the  first  applica- 
tion this  is  to  be  written  in  the  form 

^=^.330' 

243^243* 

for  which  n—\  and  ^  =  —  2.     To  make  the  last  term  unity 

330 
place  x  =  — -Vy  and  the  equation  becomes 

whence  a  =  33oy3.243'.  These  values  of  n,  k,  and  a  are  mow 
inserted  in  the  above  general  value  of  y,  and  00  made  unity: 
thus  ^=0.9983,  whence  ;irj=  1.368  is  one  of  the  roots.  For 
the  second  application  the  equation  is  to  be  written 

x'=~  '^x-'  +  243, 

243     ^  ^^' 


Art.  15.]  CONCLUSION.  31 

for  which  «  =  2  and  k=  i.     Placing  x  —  243*^,  this  becomes 

•^  2433-^     ^ 

whence  «  =  —  1 10/243',  and  the  series  is  convergent.  These 
values  of  n^  ky  and  a  are  now  inserted  in  the  formula  for  y, 
and  00  is  made  +  i  and  —  i  in  succession,  thus  giving  two 
values  for  j^,  from  which  x^  =  14.86  and  jr,  =  —  16.22  are  the 
other  roots  of  the  given  cubic. 

McClintock  has  also  given  a  similar  and  more  general 
method  applicable  to  other  algebraic  equations  than  trinomials. 
The  equation  is  reduced  to  the  form  y""  =  na  .  (I>y  ±.  i,  tvhere 
na .  (py  denotes  all  the  terms  except  the  first  and  the  last. 
Then  the  values  of  y  are  expressed  by  the  series 


+('^""£)'<"-"(«^'^)'-3l  + 


in  which  the  values  of  go  are  to  be  taken  as  before.  The 
method  is  one  of  great  importance  in  the  theory  of  equations, 
as  it  enables  not  only  the  number  of  real  and  imaginary  roots 
to  be  determined,  but  also  gives  their  values  when  the  conver- 
gence of  the  series  is  secured. 

Prob.  20.  Compute  by  the  above  method  all  the  roots  of  the 
quartic  x*  -\-  x  -\-  10  =^  o. 

Art.  15.    Conclusion. 

While  this  Chapter  forms  a  supplement  to  the  theory  of 
equations  as  commonly  given  in  college  text-books,  yet  the 
brief  space  allotted  to  it  has  prevented  the  discussion  and  de- 
velopment of  many  interesting  branches.  Chief  among  these 
is  the  topic  of  complex  or  imaginary  roots,  particularly  of 
their  graphical  representation  and  their  numerical  computation. 
Although  such  roots  rarely,  if  ever,  are  required  in  the  solution 
of  problems  in  physical  science,  their  determination  is  a  matter 
of  much  theoretic  interest.     It  may  be  mentioned,  however, 


32  THE    SOLUTION    OF    EQUATIONS.  [ChAP.  L 

that  both  the  regula  falsi  and  Newton's  approximation  rule 
may,  by  a  slight  modification,  be  adapted  to  the  computation 
of  these  imaginary  roots,  approximate  values  of  them  being 
first  obtained  by  trial. 

A  method  of  solution  of  numerical  algebraic  equations, 
which  may  be  called  a  logarithmic  process,  was  published  by 
Graffe  in  1837,  and  exemplified  by  Encke  in  1841.*  It  consists 
in  deriving  from  the  given  equation  another  equation  whose 
roots  are  high  powers  of  those  of  the  given  one,  the  coefficients 
of  the  latter  then  easily  furnishing  the  real  roots  and  the 
modttluses  of  the  imaginary  roots.  •  The  method,  although 
little  known,  is  without  doubt  one  of  high  practical  values,  as 
logarithmic  tables  are  used  throughout;  moreover,  Encke  states 
that  the  time  required  to  completely  solve  an  equation  of 
the  seventh  degree  with  six  imaginary  roots,  as  accurately  as 
can  be  done  with  seven-place  tables,  is  less  than  three  hours. 

The  algebraic  solutions  of  the  quadratic,  cubic,  and  quartic 
equations  are  valid  not  only  for  real  coefficients,  but  also  for 
imaginary  ones.  In  the  latter  case  the  imaginary  roots  do  not 
necessarily  occur  in  pairs.  The  method  of  McClintock  has  the 
great  merit  that  it  is  applicable  also  to  equations  with  imagi- 
nary coefficients;  it  constitutes  indeed  the  only  general  method 
by  which  the  roots  in  sqch  cases  can  be  computed. 

Prob.  21.  Compute  by  McClintock's  series  the  roots  of  the  equa- 
tion x^  —  i'x  —  I  =  o. 

Prob.  22.  Solve  the  equation  cos  ^  cosh  ^  +1  =  o,  and  also  the 
equation  x  —  e''  =  o.  (For  answers  see  Crelle's  Journal  fiir  Mathe- 
matik,  1841,  Vol.  XXII,  pp.  1-62.) 

*  See  Crelle's  Journal  fur  Mathematik,  1841,  pp.  193-248, 


Art.  1.]  INTRODUCTION.  33 


Chapter  II. 

DETERMINANTS. 

By  Laenas  Gifford  Weld, 
Professor  of  Mathematics  in  the  State  University  of  Iowa. 

Art.  1.    Introduction. 

As  early  as  1693  Leibnitz  arrived  at  some  vague  notions 
regarding  the  functions  which  we  now  know  as  determinants. 
His  researches  in  this  subject,  the  first  account  of  which  is 
contained  in  his  correspondence  with  De  L'Hospital,  resulted 
simply  in  the  statement  of  some  rather  clumsy  rules  for  elimi- 
nating the  unknowns  from  systems  of  linear  equations,  and 
exerted  no  influence  whatever  upon  subsequent  investigations 
in  the  same  direction.  It  was  over  half  a  century  later,  irk 
1750,  that  Gabriel  Cramer  first  formulated  an  intelligible  and 
general  definition  of  the  functions,  based  upon  the  recognition 
of  the  two  classes  of  permutations,  as  presently  to  be  set  forth. 

Though  Cramer  failed  to  recognize,  even  to  the  same  extent 
as  Leibnitz,  the  importance  of  the  functions  thus  defined,  the 
development  of  the  subject  from  this  time  on  has  been  almost 
continuous  and  often  rapid.  The  name  "  determinant"  is  due 
to  Gauss,  who,  with  Vandermonde,  Lagrange,  Cauchy,  Jacobi, 
and  others,  ranks  among  the  great  pioneers  in  this  development. 

Within  recent  years  the  theory  of  determinants  has  come 
into  very  general  use,  and  has,  in  the  hands  of  such  mathema- 
ticians as  Cayley  and  Sylvester,  led  to  results  of  the  greatest 
interest  and  importance,  both  through  the  study  of  special 
forms  of  the  functions  themselves  and  through  their  applica- 
tions.* 

*  A  list  of  writings  on  Determinants  is  given  by  Muir  in  Quarterly  Journal 
of  Mathematics,  1881,  Vol.  XVIII,  pp.  iro-149. 


34  DETERxMINANTS.  [ChAP,  II. 

Art.  2.    Permutations. 

The  various  orders  in  which  the  elements  of  a  group  may- 
be arranged  in  a  row  are  called  their  permutations. 

Any  two  elements,  as  a  and  b,  may  be  arranged  in  two 
orders  :  ab  and  ba.  A  third,  as  <;,  may  be  introduced  into  each 
of  these  two  permutations  in  three  ways  :  before  either  element, 
or  after  both  ;  thus  giving  3x2  =  6  permutations  of  the  three 
elements.  In  like  manner  an  additional  element  may  be  intro- 
duced into  each  of  the  permutations  of  i  elements  in  {i -\-  i) 
ways:  before  any  one  of  them,  or  after  all.  Hence,  in 
general,  if  Pi  denote  the  number  of  permutations  'of  i  ele- 
ments, P,^,  =  {i-\-  \)Pi.  Now,  P3  =  3X2X1=3!;  hence 
P^  =  4  X  3  !  =  4 ! ;  and,  n  being  any  integer, 

P^  =  n{n  —  \){n  —  2) . . .  i  z=  n\. 
That  is,  the  number  of  permutations  of  n  elements  is  nl. 

For  all  integral  values  of  n  greater  than  unity,  nl  is  an 
even  number. 

If  the  elements  of  any  group  be  represented  by  the  differ- 
ent letters,  ^,  b,  Cj . .  .,  the  alphabetical  order  will  be  considered 
as  the  natural  order  of  the  elements.  If  represented  by  the 
same  letter  with  different  indices,  thus : 

^j,  ^„  ^3, . . . ;     or  thus :     a\  a" ,  a"\  . . ., 

the  natural  order  of  the  elements  is  that  in  which  the  indices 
form  a  continually  increasing  series. 

Any  two  elements,  whether  adjacent  or  not,  standing  in 
their  natural  order  in  a  permutation  constitute  a  permanence ; 
standing  in  an  order  which  is  the  reverse  of  the  natural,  an 
inversion.  Thus,  in  the  permutation  daecb^  the  permanences 
are  de,  ae,  ab,  ac ;  the  inversions,  da,  dc,  db^  ec,  eb,  cb. 

The  permutations  of  the  elements  of  a  group  are  divided 
into  two  classes,  viz.:  even  or  positive  permutations,  in  which 
the  number  of  inversions  is  even  ;  and  odd  or  negative  permu- 
tations, in  which  the  number  of  inversions  is  odd. 


Art.  3.]  INTERCHANGE    OF    TWO    ELEMENTS.  35 

When  the  elements  are  arranged  in  the  natural  order  the 
number  of  inversions  is  zero — an  even  number. 

Thus,  the  even  or  positive  permutations  of  the  elements 
a^,  a^,  ^3  are 

while  the  odd  or  negative  permutations  are 


Art.  3.    Interchange  of  Two  Elements. 

It  will  now  be  shown  that  if,  in  any  permutation  of  the 
elements  of  a  group,  two  of  the  elements  be  interchanged  the 
class  of  the  permutation  will  be  changed. 

Let  q  and  s  be  the  elements  in  question.  Then,  represent- 
ing collectively  all  the  elements  which  precede  these  two  by 
P,  those  which  fall  between  them  by  Ry  and  those  which  follow 
by  7",  any  permutation  of  the  group  may  be  written 

PqRsT, 
Of  the  elements  R,  supposed  to  be  r  in  number,  let  represent 

h  the  number  of  an  order  higher  than  q, 

i    "         "  "    "        "      lower      "     q, 

j    "         '*  "    "       "      lower      "     s, 

k     "         "  "    "       "      higher    "     s. 

It  is  evident  that  no  change  in  the  order  of  the  elements  qRs 

can  affect  their  relations  to  the   elements   of   either  P  or  T. 

Then,  passing  from  the  order  PqRsT  to  the  order 

PRqsT 

changes  the  number  of  inversions  by  {h  —  i)  ;  and  passing  from 

this  to  the  order 

PsRqT 

again  changes  the  number  of  inversions  by  (j  —  k)  ±  I,  the 

\   p!"^    I  sign  being  used  as  q  is   of  \  i?^^^^  I  order   than   s, 
(  mmus  (     &  &  (  higher  j 

The  total  change  in  the  number  of  inversions  due  to  the  inter- 
change of  the  two  elements  in  question  is,  therefore, 

//  —  2 +y — ^  ±  I. 


36  DETERMINANTS.  [ChaP.  IL 

But  since  i  =  r  —  h  and  k  =z  r  —j\  this  may  be  written 

2(>^+y-  r)  ±  I, 
which  is  an  odd  number  for  all  admissible  values  of  //,/,  and  r. 
Hence,  the  interchange  of  any  two  elements  in  a  permutation 
changes  the    number  of   inversions  by  an    odd   number,  thus 
changing  the  class  of  the  permutation. 

Art.  4.    Positive  and  Negative  Permutations. 

Of  all  the  permutations  of  the  elements  of  a  group,  one 
half  are  even  and  one  half  odd. 

To  prove  this,  write  out  all  the  permutations.  Now  choose 
any  two  of  the  elements  and  interchange  them  in  each  permu- 
tation.    The  result  will  be  the  same  set  of   permutations  as 

(  even  ) 
before,  only  differently  arranged.     But  each  j  ^^^  >  permuta- 
tion of  the  old  set  has  been  converted  into  an  ]  pypj,  [  one  in 
the  new.     Hence,  in  either  set,  there  are  as  many  even  permu- 
tations as  odd ;  that  is,  one  half  are  even  and  one  half  odd. 
Prob.  I.  Classify  the  following  permutations: 
(i)  bcdea  ;  (2)  iii  v  i  11  iv;  (3)  knitnlj\ 

(4)  a"a^a'a^^a'"',  (s)  /SeyQaS;  (6)  52413; 

(7)  x^x^x^x^x.x^;  (8)   F.  Tu.  M.  Th.  W.;  (9)  jukviX, 

Prob.  2.  Derive  the  formula  for  the  number  of  permutations  of 
n  elements  taken  m  at  a  time.     (Ans.  nl/(n  —  m)\.) 

Prob.  3.  How  many  combinations  of  m  elements  arranged  in  the 
natural  order  may  be  selected  from  a  group  of  n  elements?  (Ans. 
n\/m\{n  —  m)\?) 

Prob.  4.  Show  that  o!  =  i. 

Art.  5.    The  Determinant  Array. 

Assume  n^  elements  arranged  in  n  vertical  ranks  or  columns,, 
and  n  horizontal  ranks  or  rows,  thus : 
a;  a;' . . .  «,(-) 
a:  a:' . . .  ^,(«) 


a"        a  (") 


Art.  6.]       determinant  as  function  of  «'  elements.  37 

In  this  array  all  the  elements  in  the  same  column  have  the 
same  superscript,  and  those  in  the  same  row  the  same  subscript. 
The  columns  being  arranged  in  order  from  left  to  right,  and 
the  rows  likewise  in  order  from  the  top  row  downward,  the 
position  of  any  element  of  the  array  is  shown  at  once  by  its 
indices.  Thus,  a^"  is  in  the  third  column  and  the  fifth  row 
of  the  above  array. 

The  diagonal  passing  through  the  elements  a^ ^  ^/', . .  .  a^'^^ 
is  called  the  principal  diagonal  of  the  array  ;  that  passing 
through  ^/,  ««_i'', . .  .  <3;/"\  the  secondary  diagonal.  The  posi- 
tion occupied  by  the  element  a^  is  designated  as  the  leading 
position. 

Art.  6.    Determinant  as  Function  of  n^  Elements. 

The  array  just  considered,  inclosed  between  two  vertical 
bars,  thus  : 


a'aJ'.. 

.  <^„'"> 

is  used  in  analysis  to  represent  a  certain  function  of  its  n^  ele- 
ments called  their  determinant.*  This  function  may  be  defined 
as  follows : 

Write  down  the  product  of  the  elements  on  the  principal 
diagonal,  taking  them  in  the  natural  order ;  thus  : 

This  product  is  called  the  principal  term  of  the  determinant. 
Now  permute  the  subscripts  in  this  principal  term  in  every 
possible  way,  leaving  the  superscripts  undisturbed.  To  such  of 
the  n !  resulting  terms  as  involve  the  even  permutations  of  the 
subscripts  give  the  positive  sign ;  to  those  involving  the  odd 

*This   notation    was   first   employed   by  Cauchy   in    1815.      See    Dostor's 
Th^orie  des  determinants,  Paris,  1877. 


38 


DETERMINANTS. 


[Chap.  IL 


permutations,  the  negative  sign.  The  algebraic  sum  of  all  the 
terms  thus  formed  is  the  determinant  represented  by  the 
given  array. 


Art.  7.     Examples  of  Determinants. 

Applying  the  process  above   explained  to  the  array  of  four 
ments  sfives 


elements  gives 


<  < 


~a;a:'  -  a;a;\ 


(I) 


As  an  example  of  a  determinant  of  nine  elements,  with  its  ex- 
pansion, may  be  written 


a;  a;'  a;" 
a;  a;'  a:" 
a;  a:'  a;" 


I      ^  /^  n ^  ///     I      ^  t ^  11  ^  "f     I      ^  r ^  n ^  r 

^iz  -\-  a^a^  a^     -\-  a^  a^  a^      -j-  a^  a^  a^ 


(-') 


It  is  evident,  from  the  mode  of  its  formation,  that  each  term 
of  the  expansion  of  a  determinant  contains  one,  and  only  one^ 
element  from  each  column  and  each  row  of  the  array. 

It  follows  that  every  complete  determinant  is  a  homoge- 
neous function  of  its  elements.  The  degree  of  this  function, 
with  respect  to  its  elements,  is  called  the  order  of  the  deter- 
minant. Thus,  (i)  and  (2)  are  of  the  second  and  third  order 
respectively. 

The  definition  of  a  determinant  given  in  the  preceding 
article  is  once  more  illustrated  by  the  following  example  of  a 
determinant  of  the  fourth  order  with  its  complete  development : 


a^  b^  <:,  d^ 

a,  b^  c,  < 

a^  b^  c^  d^ 

a^  b^^  c^  d^ 


+  aj)^c,d^  —  afi.c^d^  —  a^b.c^d^  +  a^b.c^d^ 
-\-  aj}^c^d^  —  afi^c^d^  —  aj)^c^d^  -\-  ajb^c^d.^ 
-\-  a.J)^c4,  —  afi.c^^  —  aj)^c^d^  +  ^,V,^3 
+  aj),c^d^  —  aj)^c4^  —  a^b,c,d^  +  a,b,c,d^ 


>  (3) 


Art.  8.] 


NOTATIONS. 


39 


It  will  be  noticed  that,  in  this  case,  the  columns  are  ranked 
alphabetically  instead  of  by  the  numerical  values  of  a  series  of 
indices. 


Art.  8.    Notations. 

Besides  the  notations  already  employed,  the  following  is 
very  extensively  used  : 


^n 

ci^,  . 

.  .a,^ 

a.. 

a^,  . 

•  .^,« 

a„ 

.  ^n. 

•   .    ilnn 

This  is  called  the  double-subscript  notation  ;  the  first  subscript 
indicating  the  rank  of  the  row,  the  second  that  of  the  column. 
Thus  the  element  ^„  is  in  the  second  row  and  the  third  column. 
The  letters  are  sometimes  omitted,  the  elements  being  thus 
represented  by  the  double  subscripts  alone.* 

Instead  of  writing  out  the  array  in  full,  it  is  customary, 
when  the  elements  are  merely  symbolic,  to  write  only  the  prin- 
cipal term  and  enclose  it  between  vertical  bars.  This  is  called 
the  umbral  notation.  Thus,  the  determinant  of  the  nth.  order 
is  written 

I  a:  a:'  .  .  .  a.<")  |  ; 

or,  using  double  subscripts, 

I    ^n   ^«   •   .   ,  ^««    I    . 

These  last  two  forms  are  sometimes  still  further  abridged  to 


in) 


and 


respectively. 


Prob.  5-  Write  out  the  developments  of  the  following  determi- 
nants: 


a,  d, 

;     (2) 

//' 

;     (3) 

p'    q' 

;     (4) 

a   b 

<i,K 

//' 

/'  /' 

«/} 

(l) 


*  Leibnitz  indicated  the  elements  of  a  determinant  in  this  same  manner, 
though  he  made  no  use  of  the  array. 


40 

(5)  I  <^//3 1  ;  (6) 


DETERMINANTS. 


[Chap.  II. 


/     x//    j.fff 


p  p  p 


;  (7) 


f    q' 


p"  q"  ^" 


;    (8) 


(9)  I  II.  22  I  ;     (10) 


a  b  c 

X  y  z 

;ii)  I  l,m,n^  I  ;     (12)  |  a^^a^^a,,a,^  \  . 

Prob.  6.  How  many  terms  are  there  in  the  development  of  the 
determinant  |  aj^  \  ? 

In  the  above  determinant  tell  the  signs  of  the  terms  : 

(i)  a/a^ar^r^^/".  (2)  ^r'<'<''<W<; 

(3)  a/arar<W<'' 

Prob.  7.  Show  that  in  the  expansion  of  any  determinant,  all  of 
whose  elements  are  positive,  one  half  the  terms  are  positive  and  one 
half  negative. 

Prob.  8.  In  determinants  of  what  orders  is  the  term  containing 
the  elements  on  the  secondary  diagonal  (called  the  secondary  term) 
positive  ? 

Prob.  9.  What  is  the  order  of  the  determinant  whose  secondary 
term  contains  10  inversions  .'*  36  inversions  ? 

Prob.  10.  In  the  expansion  of  a  determinant  of  the  nth.  order, 
how  many  terms  contain  the  leading  element  ? 


Art.  9.    Second  and  Third  Orders. 

Simple  rules  will  now  be  given  for  writing  out  the  expan- 
sions of  determinants  of  the  second  and  third  orders  directly 
from  the  arrays  by  which  they  are  represented. 

To  expand  a  determinant  of  the  second  order,  write  the 
product  of  the  elements  on  the  principal  diagonal  minus  the 
product  of  those  on  the  secondary  diagonal,  thus : 

=  ad  —  be, 

=  -3  + 

The  following  method  is  applicable  to  determinants  of  the 
third  order  :* 

*  This  method  was  first  given  by  Sarrus,  and  is  often  called  the  rule  of  Sarrus; 
sec  Finck's  Elements  d'Algebre,  1846,  p.  95. 


Likewise, 


a     b 

c     d 

-9    5 

2     il 

=  -  3  + 10  =  7. 


Art.  9.] 


SECOND    AND    THIRD    ORDERS. 


41 


Beneath  the  square  array  let  the  first  two  rows  be  repeated  in 
order,  as  shown  in  the  figure. 
Now  write  down  six  terms,  each 
the  product  of  the  three  ele- 
ments lying  along  one  of  the 
six  oblique  lines  parallel  to  the 
diagonals  of  the  original  square. 
Give  to  those  terms  whose  ele- 
ments lie  on  lines  parallel   to  __ 
the  principal  diagonal  the  posi- 
tive  sign;   to  the   others,  the  — 
negative  sign.     The   result   is 
the  required  expansion.     Ap- 
plying  the  method  to  the  determinant  just  written  gives 

I  ^1  Vs  I  =  ^i<^»<^3  +  '^aVi  +  cij),c^  —  aj).f,  —  a,b,c^  —  aj),c^. 
After  a  little  practice  the  repetition  of  the  first  two  rows  will 
be  dispensed  with. 

The  above  methods  are  especially  useful  in  expanding 
determinants  whose  elements  are  not  marked  with  indices,  or 
in  evaluating  those  having  numerical  elements.  No  such  sim- 
ple methods  can  be  given  for  developing  determinants  of  higher 
orders,  but  it  will  be  shown  later  that  these  can  always  be 
resolved  into  determinants  of  the  third  or  second  order. 

Prob.  II.  Develop  the  following  determinants: 


(i) 


(4) 


(7) 


a  h  g 
h  b  f 
S  f  c 


(2) 


^.y.  I 

» 

(5) 

x,y,  I 

^^y-.  I 

I      cos  a 

;      (8) 

cos  a 

I 

o  —  n  — m 

) 

(3) 

A   c 

b 

> 

n      o  —  / 

c    Ba 

m      I      o 

b    a  C 

I      P        Q 

;      (6) 

cos  a  sin  ^ 

o  cos  a  sin  /S 

sin  a  cos  fi 

0  sin  a  cos  fi 

I   /_  I 

> 

(9) 

a  b  c 

. 

4    V-2 

cab 
b  c  a 

(l) 


Prob.  12.  Evaluate  the  following: 


1  2    3 

3   I   2 

2  3    I 


[2) 


-  2 

O 

12 


2  4 

2    O 
2     I 


(3) 


_  I  -  i/^ri  -  s/  -  1 

(Ans.  i8;   i6;  2.) 


42 


DETERMINANTS. 


[Chap.  II. 


Art.  10.    Interchange  of  Rows  and  Columns. 
Any  term  in  the  development  of  the  determinant  |  ^Z")]  may- 
be written 

±^  I   ^  tt   ^  itr  ^  in) 

in  which  ^2/'. .  ./is  some  permutation  of  the  subscripts  i,  2,  3,.. .«. 
Designate  by  u  the  number  of  inversions  in  hij . .  .  /.  Also,  let 
V  be  the  number  of  interchanges  of  two  elements  necessary  to 
bring  the  given  term  into  the  form 


±^, 


{i>) 


{q)  a  (^) 


.  a 


it) 


in  which  the  subscripts  are  arranged  in  the  natural  order,  while 
pqr . . .  /  is  a  certain  permutation  of  the  superscripts  ',  '\  "\  . . .  ^"^ 

This  permutation  is  even  or  odd  according  as  v  is  even  or 
odd.  But  u  and  v  are  obviously  of  the  same  class ;  that  is, 
both  are  even  or  both  odd.  Hence  the  permutations  hij . . .  / 
and  pgr  . . .  /  are  of  the  same  class ;  and  the  term  will  have  the 
same  sign,  whether  the  sign  be  determined  by  the  class  of  the 
permutation  of  the  subscripts  when  the  superscripts  stand  in 
the  natural  order,  or  by  the  class  of  the  permutation  of  the 
superscripts  when  the  order  of  the  subscripts  is  natural. 

It  follows  that  the  same  development  of  the  determinant 
array  will  be  obtained  if,  instead  of  proceeding  as  indicated  in 
Art.  6,  the  superscripts  of  the  principal  term  be  permuted,  the 
subscripts  being  left  in  the  natural  order,  and  the  sign  of  each 
of  the  resulting  terms  written  in  accordance  with  the  class  of 
the  permutations  of  its  superscripts. 

Passing  from  one  of  these  methods  of  development  to  the 
other  amounts  to  the  same  thing  as  changing  each  column  of 
the  array  into  a  row  of  the  same  rank,  and  vice  versa.  Hence, 
a  determinant  is  not  altered  by  changing  the  columns  into  cor- 
responding rows  and  the  rows  into  corresponding  columns. 
Thus: 


a'  a. 


.  .a, 
.  .  a. 


(«) 


(«) 


^n    Cln 


a^^  I 


.  .  a. 


(«) 


Art.  11.]  TWO    IDENTICAL    PARALLEL    LINES.  43 

Whatever  theorem,  therefore,  is  demonstrated  with  reference 
to  the  rows  of  a  determinant  is  also  true  with  reference  to  the 
columns. 

The  rows  and  columns  of  a  determinant  array  are  aHke 
called  lines. 

Art.  11.    Interchange  of  Two  Parallel  Lines. 

If  any  two  parallel  lines  of  a  determinant  be  interchanged, 
the  determinant  will  be  changed  only  in  sign. 

For,  interchanging  any  two  parallel  lines  of  a  determinant 
array  amounts  to  the  same  thing  as  interchanging,  in  every 
term  of  the  expansion,  the  indices  which  correspond  to  these 
lines.  Since  this  changes  the  class  of  each  permutation  of  the 
indices  in  question  from  odd  to  even  or  from  even  to  odd,  it 
changes  the  sign  of  each  term  of  the  expansion,  and  therefore 
that  of  the  whole  determinant. 

It  follows  from  the  above  that  if  any  line  of  a  determinant 
be  passed  over  m  parallel  lines  to  a  new  position  in  the  array 
the  new  determinant  will  be  equal  to  the  original  one  multi- 
plied by  (—  i)'". 

The  element  Uk^'^  may  be  brought  to  the  leading  position 
by  passing  the  >^th  row  over  the  {k  —  i)  preceding  rows,  and 
the  .yth  column  over  the  {s —  i)  preceding  columns.  This 
being  done  the  determinant  is  multiplied  by 

(- 1)*- .  (- 1)'-  =  (-  i)'n 

which  changes  its  sign  or  not  according  as  (k-\-s)  is  odd  or 
even. 

The  position  occupied  by  a^''^  is  called  a  positive  position 
when  {k  +  s)  is  even  ;  a  negative  position  when  {k  -\-  s)  is  odd. 

Art.  12.    Two  Identical  Parallel  Lines. 

A  determinant  in  which  any  two  parallel  lines  are  identical 
is  equal  to  zero. 

For  the  interchange  of   these  two  parallel  lines,  while  it 


44 


DETERMINANTS. 


[Chap.  IL 


changes  the  sign  of  the  determinant,  will  in  no  way  alter  its 
value.     The  value  then,  if  finite,  can  only  be  zero. 

Art.  13.    Multiplying  by  a  Factor. 

Multiplying  each  element  of  a  Hne  of  a  determinant  by  a 
given  factor  multiplies  the  determinant  by  that  factor. 

Since  each  term  of  the  development  contains  one  and  only 
one  element  from  the  line  in  question  (Art.  7),  then  multiply- 
ing each  element  of  this  line  by  the  given  factor  multiplies 
each  term  of  the  development,  and  therefore  the  whole  deter- 
minant, by  the  same  factor. 

It  follows  that,  if  the  elements  of  any  line  of  a  determinant 
contain  a  common  factor,  this  factor  may  be  canceled  and  written 
outside  the  array  as  a  factor  of  the  whole  determinant ;  thus  : 


a^^.  .  m  a^i 


a,. 


=  m  \  a^^  a^ 


.  a^ 


A  determinant  in  which  the  elements  of  any  line  have  a 
common  ratio  to  the  corresponding  elements  of  any  parallel 
line  is  equal  to  zero.  For  this  common  ratio  may  be  written 
outside  the  array,  which  will  then  have  two  identical  lines.  Its 
value  is  therefore  zero  (Art.  12). 

A  determinant  having  a  line  of  zeros  is  equal  to  zero. 


Art.  14.    A  Line  of  Polynomial  Elements. 

A  determinant  having  a  line  of  elements  each  of  which  is 
the  sum  of  two  or  more  quantities  can  be  expressed  as  the 
sum  of  two  or  more  determinants. 


Let       a,     {b,-b:-^rb:'  ±.,.)     c,  ,  .  . 


be  such  a  determinant.     Then,  if 

B,~b,-b!-\-b!'  ± 


=  A 


(I) 


Art.  15.]  COMPOSITION    OF    PARALLEL    LINES.  45- 

any  term  of  the  expansion  of  the  determinant  A  is 

±  Uf,  BiCj  .  .  .  =  ±a;,  diCy.  .  .  ^^  a^  d/  9 .  .  . 

±  a,  bl'  ^y .  .  .  ±  .  .  .  (2> 

The  terms  in  the  expan^^n  of  A  are  obtained  by  permuting 
the  subscripts  h,  i,j\ .  .  .of  aj,  Bi  Cj  .  .  ,  .  But  permuting  at 
the  same  time  the  subscripts  of  the  terms  in  the  second  mem- 
ber of  (2),  and  giving  to  each  term  thus  obtained  its  proper 
sign,  there  results 

which  proves  the  theorem. 

Art.  15.    Composition  of  Parallel  Lines. 

If  each  element  of  a  line  of  a  determinant  be  multiplied  by 
a  given  factor  and  the  product  added  to  the  corresponding  ele- 
ment of  any  parallel  line,  the  value  of  the  determinant  will  not 
be  changed  ;  thus: 

a,,  a,^  a,,.  .  .  a,„     =    a,,  a,^  {a,,  +  ma,,)  .  .  ,  a,^\ 

^«i  ^m  ^«3  •   •   •  ^nn  ^«i  ^«2  (^«3  +  ^^mi)  •   •   •  ^«« 

This  wi!l  appear  upon  resolving  the  second  member  into 
two  determinants  (Art.  14),  one  of  which  will  be  the  given  de- 
terminant, while  the  other,  upon  removal  of  the  given  factor, 
will  vanish  because  of  having  two  identical  lines. 

In  like  manner  any  number  of  parallel  lines  may  be  com- 
bined without  changing  the  value  of  the  determinant,  care 
being  taken  not  to  modify  in  any  way  the  elements  to  which 
are  added  multiples  of  corresponding  elements  from  other 
parallel  lines.     For  example,  |  ^,,„  |  is  equivalent  to 

^11  (^^u  +  ^12  —  ^^^.3  +  •  •  •)      a,,  ...  a,^ 

-;«(^,3  +  A^,3)-f.  .  .) 
{la^,  +  ^„,  —  ma„,  +  .  . .)     ^„8  •  •  •  ^nn 


46 


DETERMINANTS. 

Art.  16.     Binomial  Factors. 


[Chap.  II. 


A  determinant  which  is  a  rational  integral  function  of  a 
and  of  b,  such  that  if  b  is  substituted  for  a  the  determinant 
vanishes,  contains  {a  —  <5)  as  a  facton     For  example, 

A^  a"  —  p"^   a  —  q     a-\-r 
b'-f    b-q     b  +  r 
p  q  r 

is  divisible  by  {a  —  b). 

To  prove  this,  let  the  expansion  of  any  such  determinant 
be  written  in  the  form 

J  =  nt^-\-  m^a  -\-  m^d"  + . . . , 

the  coefficients  m^,  m^,  m^,  .  .  .  being  independent  of  a.      Now 
when  b  is  substituted  for  a  the  determinant  vanishes.     Hence, 

o  z=  m^-^-  mjb  -\-  mj}^  -j-  .  .  . 

Subtracting  this  from  the  preceding  gives 

A  =  m^a  -b)-\-  m^a^  _  ^»)  -|-  .  .  . 

This  being  divisible  by  {a  —  b),  the  theorem  is  proven. 

Prob.  13.  Prove  the  following  without  expansion  : 


(i) 


0      —X       X 

=  0; 

(2) 

0 

c 

-  b 

my      0    —y 

—  c       0 

a 

—  mnz  nz      0 

b  -a 

0 

(3) 

b  -{-  c      a         a 

b      c  -Y  a      b 
c          c       a  -\-  b 

=   2 

0     c     b 
c    0     a 
b    a     0 

' 

(4) 

b'  -}-c' 

—       a            a 

a 

*      b      ^ 

=    2 

0     c     b 
c    0     a 
b    a     0 

» 

c             c 

-  b' 

=  o; 


(S) 


a  sin^  b  —  c 
b  sin  j5  c  —  a 
c     sin  C      a  —  b 


—  o,  the  elements  referring  to   the 
triangle  ABC. 


Art.  17.] 


CO-FACTORS  :     MINORS. 


47 


I  X  y 

IX  y 

o  x,—x  y,—y 
ox^-x  y^-y 


Prob.  14.  Prove  that 

I  X  —a  y  —b 
1  x,—a  y^—b 
ix^  —  a  y^  —  b 

Prob.  15.  Find  the  value  of  d  in  the  equation 

sin  /^  sin  6^     o         =0.     (Ans.  0  =  tt/^.) 

I  o  I 

o     cos  0  cos  0 

Prob.  16.  Show  that  the  proportion  a:b::/:m  may  be  written 
in  the  form  .  =0;  and  from  the  properties  of  this  determinant 
prove  the  common  theorems  in  proportion. 

Prob.  17.  Show  that  the  determinant 

factor  {be  -{-  ca  +  ab). 

Prob.  18.  Resolve  the  following  determinants  into  factors:* 

(i) 


ab  c'  ^' 

a'  be  a' 

contains  the 

b"  b'  ca 

la  a"^ 

» 

{-) 

la  a^ 

a' 

» 

(3) 

I  a,  a 

"  "    'd. 

n- 

lb  b"" 

lb  b'  b' 

la^a^  ..  .  a/- 

ic  c" 

le  e'  e' 
I  d  d'  d' 

lanOn     .    .    .   «„"" 

I      I     I      T 

i 

(5) 

I    I    I 

;       (6) 

Ill 

, 

abed 

a  b    e 

a  b   e 

a'  b'  c'  d-^ 

a'  b'  c* 

a'  b'  c* 

a*  b* 

c'd* 

(4) 


Art.  17.    Co-factors;  Minors. 

The  terms  of  J  e  |  ^/"^l  which  contain  the  element  a/  may- 
be obtained  by  expanding  the  determinant 

(I) 


^/  0    0 

.  .0 

a„'  a^'  a^"  . 

For,  in  writing  out  this  expansion  each  term  is  formed  by 
taking  one,  and  only  one,  element  from  each  column  and  each 

*  These  determinants  belong  to  an  important  class  known  as  alternates. 
See  Hanus'  Elements  of  Determinants,  Boston,  1888,  pp.  187-201. 


48 


DETERMINANTS, 


[Chap.  IL 


row  of  the  array  (Art.  7).  If,  therefore,  in  selecting  the  ele- 
ments for  any  term,  any  other  element  than  a^  be  taken  from 
the  first  column,  the  one  taken  from  the  first  row  must  be  zero. 
Hence,  the  only  terms  which  do  not  vanish  are  those  which 
contain  the  element  a^. 

Moreover,  in  the  terms  of  the  expansion  of  (i)  which  do 
not  vanish,  a^  is  multiplied  by  {n  —  i)  elements  chosen  one 
from  each  column  and  each  row  of 

(2) 


(«) 


There  are  {n  —  i)!  such  terms,  any  one  of  which  may  be 
written  ±  a^al'aj"  .  .  .  ^/"^ ;  the  sign  being  determined  by  the 
class  of  the  permutation  of  the  n  subscripts  i,  t,j\  .  .  .  /.  But 
since  this  is  of  the  same  class  as  the  permutation  of  the  {n  —  i) 
subscripts  /,/, . . .  /,  the  sign  of  any  term,  ±  a^al'aj"  . . .  <2/''\ 
of  the  expansion  of  (i)  is  the  same  as  the  sign  of  the  corre- 
sponding term,  al'a!" .,  .a/'*\  of  the  expansion  of  (2).     Hence, 

-<   a:'  a:"..M^^^  .  (3> 


a^  o 


a:"..M^-^ 


a^  a^ 


.  .a 


(«) 


^  //  ^  tn         „ 
an      Hn       .  .  .Ut 


(«) 


The  determinant  (2)  is  called  the  co-factor  or  complement 
of  the  element  a/  in  the  determinant  |  ^/"^  |.  It  is  obtained 
from  this  determinant  by  deleting  the  first  column  and  the 
first  row. 

The  co-factor  of  any  element  a^^^^  maybe  found  in  the  same 
manner  upon  transposing  this  element  to  the  leading  position. 
But  by  this  transposition  the  sign  of  the  determinant  will  be 
changed  or  not  according  as  a^'^  occupies  a  negative  or  a  posi- 
tive position  (Art.  11).  Hence,  to  find  the  co-factor  of  any 
element  ^/*^  of  the  determinant  |  ^/"^  |,  delete  the  row  and  the 
column  to  which  the  element  belongs,  giving  the  resulting 
determinant  the  |  PJ"^^^^^^^  I  sign  when  {k  +  ^)  is  |  ^^^^  I . 


Art.  18.]  development  in  terms  of  co-factors. 


49 


The  co-factor  thus  obtained  is  represented  by  the  symbol 

the  sign-factor  of  which,  (—  i)*+*,  is  intrinsic,  i.e.,  included  in 
the  symbol  itself,  which  is  accordingly  written  as  positive. 
The  co-factors  of  the  various  elements  of  |  a^^a^^a^^  \  are  as 
follows : 


^n- 

^„  ^„ 

;           ^ia=^  — 

^3.    ^„ 

;        A,,~ 

^1X    ^« 

^35    ^33 

^3,    ^33 

^31    ^3. 

^.   =   - 

^n     ^>3 

;                  -^33   = 

^11    <^13 

;        ^„  =^  — 

^n  ^:. 

<^33    ^33 

^3.    ^33 

^3,    ^3. 

^3. 

^1,     ^13 

;        ^39  —  "~ 

^n    ^,3 

;       ^33  =^ 

^n  ^.; 

^3,     ^23 

^ai  ^23 

^31     ^.. 

The  result  obtained  by  deleting  the  ^th  row  and  the  .fth 
column  of  J  ^  ]  ^^^"^  |  is  called  the  minor  of  the  determinant 
with  respect  to  the  element  ^|fe*\  and  is  written  A%.  This  minor 
is  the  same  as  the  co-factor  of  the  same  element  without  its 

sign-factor ;  thus : 

^,w  =  (-  if^^A%. 

Similarly  A\l\\  is  the  result  obtained  by  deleting  the  ^th  and 
^th  rows  and  the  /th  and  .yth  columns  of  A,  and  is  called 
a  second  minor  of  the  given  determinant.  Minors  of  still 
lower  orders  are  obtained  in  a  similar  manner,  and  expressed 
by  a  similar  notation.  The  >^th  minors  are  determinants  of 
the  order  {71  —  k). 


Art.  18.    Development  in  Terms  of  Co-factors. 

The  {n  —  i)  !  terms  of  |  ^/"^  |  which  contain  ai'"*  are  repre- 
sented in  the  aggregate  by  a^'^A^'''  (Eq.  3,  Art.  17).  In  like 
manner  the  groups  of  terms  containing  the  successive  elements 
^/»  ^a"»  .  .  •  o^i'"''  are  respectively 

a^A:,     a^'A^',  .  . .  a^^^A^-\ 

Each  one  of  these  n  groups  includes  {it  —  i) !   terms   of  the 
determinant  |  a^""^  \  ,  no  one  of  which  is  found  in    any  other 


50 


DETERMINANTS. 


[Chap,  II. 


group.  In  all  of  them,  then,  there  are  nX{n  —  i)\  or  nl  dif- 
ferent terms  of  the  determinant,  which  is  the  whole  number. 
Hence, 


(n) 


Similarly  (Art.  lo). 


(I) 


(2) 


Any  determinant  may,  by  means  of  either  (i)  or  (2),  be  re- 
solved into  determinants  of  an  order  one  lower.  Since,  in 
these  formulas  ^/,  .  .  .  ^^^"^,  or  A^^'\  .  .  .  A^'^  are  themselves 
determinants,  they  may  be  resolved  into  determinants  of  an 
order  still  one  lower  in  the  same  manner.  By  continuing  the 
process  any  determinant  may  ultimately  be  expressed  in  terms 
of  determinants  of  the  third  or  second  order,  which  may  be 
easily  expanded  by  methods  already  given  (Art.  9). 

For  example,  let  it  be  required  to  develop  the  determinant 
^  =  \  a^  b^  c^  d^  \  .     Applying  formula  (i),  letting  ^  =  i,  gives 


J  =  a. 


Upon  a  second  application  of  the  same  formula  this  becomes 


b.  c.  < 

-  b. 

a^  c^  d. 

■\-c. 

a,  b,  < 

-< 

a.  b,  c. 

b.  c,  d. 

^3  ^,  < 

a,  b,  d. 

a,  b,  c. 

K  c,  d. 

a,  c,  d^ 

a^  b^  d. 

a,  b^  c. 

=  a,K 

c,d. 

-  <^,':, 

Kd, 

+  ^.d. 

b,c. 

c,d. 

Kd, 

Kc, 

-  ".  K 

c.d. 

+  b.c. 

a,d. 

-Kd, 

a,  c. 

c,d. 

a,d. 

a,  c. 

+  «, '^, 

b,d. 

—  b,  c. 

a,d. 

+  c.d. 

a,b. 

b,d. 

a,d. 

a,  *. 

—  «,  d,    b,  c. 

+  b,d, 

a,<:. 

-c,d. 

a,b. 

b.c. 

a,c. 

a,b. 

The  complete  development  may  be  written  out  directly  from 
the  above.     It  is  given  in  Eq.  3,  Art.  7. 


Art.  19.] 


THE    ZERO    FORMULAS. 


51 


Prob,  19.  Develop  the  following  determinants: 


(i) 


I  X  I  y 
X  1  y  1 
1  y  I  X 
y  I  X  I 


2) 

a  X  y  a 

X  0  0  y 

y  0  0  X 

a  y  X  a 

(3) 


o  q  r  s 
p  o  r  s 
p  q  o  s 
p  q  r  o 


Prob.  20.  Find  the  values  of  the  following  determinants: 


(I) 


(4) 


1234 

2341 
3412 
4123 

0  I  I  I 

1  o  I  I 
I  I  O  I 
I  I  I  o 


(2) 


0102 
1020 
0201 
2010 


(3) 


5  3 

6  -I 


(5) 


-3 
3 
3333;     (6)  0003 
3222  1002 

2211  OIOI 

I  I  I  o  0010 

(Ans.  160;  9;  o;  3;  3;  3.) 

Prob.  21.  Obtain  the  determinants  in  Exs.  5  and  6  of  the  pre 
ceding  problem  from  that  in  Ex.  4. 
Prob.  22.   Evaluate 


Prob.  23.  Show  that 


01  I  . . . 

,  of  the  //th  order. 

101... 

(Ans.  («-i)(-i)«-) 

I  10... 

a 

b       c       d 

=  («'+-5'+.'+<^)' 

-  b 

a    —d       c 

—  c 

d      a    -b 

-  d   - 

c       b       a 

Art.  19.    The  Zero  Formulas. 

If  in  the  determinant  |  ^/"^  |  the  ^th  and  kt\i  rows  be  sup- 
posed identical,  the  elements  «/,  Uk' ,  .  .  .  a^'''^  in  the  formula 
(i)  of  the  last  article  may  be  replaced  by  rt:,/,  ^/',  .  .  .  a^""^  re- 
spectively. But  in  this  case  the  value  of  the  determinant  is 
zero    (Art.    12).     Hence,    in    reference    to    the    determinant 


(«) 


,  h  and  k  being  different  subscripts, 

a^A^  +  a^'A^'  + . .  .  +  a^A^"''  =  o. 


62  DETERMINANTS.  [Chap.  II. 

Similarly,  p  and  s  being  different  superscripts, 

^^(^)^^w  +  a^^^Ai^^  + .  .  •  +  ^n^^^-4«(^^  =  o. 

Art.  20.    Cauchy's  Method  of  Development. 

It  is  frequently  desirable  to  expand  a  determinant  with 
reference  to  the  elements  of  a  given  row  and  column. 

Let  the  determinant  be  //  eee  |  i^:/"^  |  ,  and  the  given  row 
and  column  the  /^th  and  p\.\i  respectively.  Then  is  Ai^'^  the 
co-factor  of  ali*^^  the  element  at  the  intersection  of  the  two 
given  lines.  The  co-factor  of  any  element  a^'''^  of  Alf>  will  be 
designated  by  Bk^'^,  this  being  a  determinant  of  the  order 
{n  —  2).  The  required  expansion  may  now  be  obtained  by 
means  of  the  following  formula,  due  to  Cauchy : 

I  «,<">  I  =  «*«'^/«-  ^ai^at*mt-\  (I) 

in  which  >^  =  i,  2, . ..  ^  —  i,  ^  -|~  ^>  •  •  •  ^>  ^"<^  ^  =  i,  2, .. ./  — i, 
/  +  !,...«,  successively. 

To  prove  this,  consider  that  B^;'^  is  the  aggregate  of  all 
terms  of  the  expansion  of  A  which  contain  the  product 
ai^'^ai'K  These  terms  are  included  in  ai^'^Ah'^\  Now,  every 
term  in  the  expansion  which  does  not  contain  a,if^  must  contain 
some  other  element  a,^''^  from  the  hth.  row  and  also  some  other 
element  a^^'^  from  the  /th  column,  and  thus  contains  the  prod- 
uct ai'^ai^\  But  this  product  differs  from  ^a^^*-*'  only  in  the 
order  of  the  superscripts  ;  and  is,  therefore,  in  the  expansion  of 
Jy  multiplied  by  an  aggregate  of  terms  differing  in  sign  only 
from  that  multiplying  a^^^a^'^.  Hence,  —  B^'^  is  the  coefficient 
of  a^^^  a^^^  in  the  required  expansion. 

In  the  formula  aj^^^A^^*^  gives  {n  —  i) !  terms  of  A,      There 
are  also  {n  —  i)'  such  aggregates  as  —  ai'^a^^^Bt^,  each    con- 
taining («  —  2) !  terms.     The  formula  therefore  gives 
(«  —  i) !  +  («  —  i)'  («  —  2) !  =  «  !  terms,  which  is  the  complete 
expansion. 

When  the  expansion  is  required  with  reference  to  the  ele- 


Ar*t.  20.] 


CAUCHY  S  METHOD  OF  DEVELOPMENT. 


53 


ments  of  the  first  column  and  the  first  row  the  formula,  written 
explicitly,  becomes 

1^,W|  =  a/ A/-  a:a;'B:'  -  a,'a,"'B,'"  -  ...  -  a^a^^^B^^^ 
-  a^al'B:'  -  a:a:"B:"  -  ...  -  «3'^/'')^3W 


-  a:a:'B:'-  a:a:"B:"-  ...  -  a:a^-^B^-\     (2) 

in  which  B^''^  has  intrinsically  the  sign  (—  I)*+^ 

Cauchy's  formula  is  particularly  useful  in  expanding  deter>- 
minants  which  have  been  bordered  ;  such  as 


u,  a,,  a,^  ^,3 
u^  a,,  a,,  a,, 

^%    ^tX    ^Sa  ^33 

Applying  formula  (2)  to  this  determinant  gives 


(3) 


^„    ^53 

+  «,«, 

^31 

^,3 

-     ^8«1 

^21    ^22 

1    ^35    ^38 

^81 

^33 

^81    ^38 

^10   ^1, 

-      «8' 

^11 

^,8 

+  «,«8 

^n   ^n 

^3a    ^38 

^31 

^,3 

^31    ^32 

^„  ^n 

+  «,«, 

^11 

^,8 

-«,' 

^n  ^>2 

a,,  «„ 

^21 

^38 

^2.    ^22 

Letting  ^^,  =  ^,4,  and  writing  ^,,,  ^„, ...  for  the  co-factors  of 
the  elements  of  |  a^^a^a^^  \  ,  the  above  becomes 

Q  =  A,,u:  +  A,,u;  +  ^33<  +  2^„«,?^3  +  2^3,?^3^^,  +  2A,,u,u,, 

Prob.  24.  Develop    the    following    determinants    by   Cauchy's 
formula: 


(i) 


a  h  g  u 
hb  fv 
gfcw 
u  vwo 


(2) 


0  yz  zx  xy 

;      (3) 

yz   0   1    I 

zx  1   0    1 

xy  1    I    0 

0  I       I      I 

1  o  xy  zx 
I  xy  o  yz 
I  zx  yz  o 


54 
(4) 


DETERMINANTS. 


—  I  —  X 

I  -y 

X  o 

1  —  z 


I       I 

;     (5) 

I    I    I  ^ 

;   (6) 

I       I 

xy  z  o 

y    z 

I  I  I JJ^ 

I  —I 

\  \  \  z 

[Chai    IL 

o  a         b 

—  a       sin  ^  sin^ 

—  b  —  cos^  cos-5 


,l^+ 


dykl 


(2) 
(3) 


Art.  21.    Differentiation  of  Determinants. 
By  the  formula  (i)  of  Art.  i8 

^  =  I  y^,n  !  =  Yk^yu.  +  y^yn  +  •  •  •  ^*«7a«.  (0 

Considering  the  elements  of  the  determinant  as  independent 

variables  and  differentiating  with  respect  to  y^,  gives 

dA 
dksA  =  Yks  dyks ,     or     F^  = 

Substituting  in  (i), 
Similarly 

Again  differentiating  (i),  this  time  with  respect  to  all  the  ele- 
ments of  the  kth.  row,  there  results 

d,A  =  Y,4yu.  +  Yk^dy,,  +  . . .  +  n«^7*«.  (5) 

In  the  total  differential  of  A-  there  are  obviously  n  such  ex- 
pressions as  (5),  each  of  which  may  be  obtained  from  A  by 
replacing  the  elements  of  some  one  of  the  rows  by  their  differ- 
entials :  thus : 


dA 


dy,. 


dy,n 

y.H 


+ 


'dy,n 


y. 


+  ...+ 


y^^ 

y.i 


y.n 
y.n 


dy„^ .  ,,dy,, 


.(6) 


If  all  the  elements  are  functions  of  one  independent  variable  ;r, 

then,  representing  —^  hy  yj, 
ax 


dA 
dx 


y.: 
y.i 


'^y.n 
"  y.n 

"  ynn 


+ 


y.^ 


'-yin 

•  yj 


+  ...+ 


yn  "'  y.n 
y,^  '  "  y.n 

yj  "-ynn 


.  (7) 


Art.  32.] 


RAISING    THE    ORDER. 


55 


Prob.  25.  Show  that  Cauchy's  formula  may  be  written 


dah^^^ 


daK^^^da^"^' 


Art.  22.     Raising  the  Order. 
Since,  in  the  expansion  of  the  determinant  (i)  of  Art.  17 
the  elements  ^/,  . . .  aj  do  not  appear,  these  may  be  replaced 
by  any  quantities  whatever,  as   Q,  . ,  .  T,  without  changing  the 
value  of  the  determinant  ;  thus: 


«/    o      o    . . .    o 

a:  a:'  a:''  .  .  .  ^^^-^ 


a:  a;'  a:"  . . .  a^""^ 


«/  o     o    . . .    o 


Similarly, 


a/    0      0     .  . 

a:  a-    0     .. 

.    0 
.    0 

=<a," 

.  ■  «,"" 

a.'". 

. .  ar 

a:  a:'  a:"  . . 

•  a„^"' 

a/  o  o  ...  o 
Qa^'  o  ...  o 
R  L  a^".  . .  ^3(''> 


in  which  Q,  Rj  . . .  T  and  L^ .  . .  N  are  any  quantities  whatever 
Finally, 

o     ...o  o    =rt,V...^«ir'W^=  ^/  o   ...o      o 

o  Q  a^".  . .  o      Q 


a       a  "  a  <""'>  a  ^"^ 


T  N...C  a^^^ 


that  is,  if  all  the  elements  on  one  side  of  the  principal  diagonal 
are  zeros  the  determinant  is  equal  to  its  principal  term,  and 
the  elements  on  the  other  side  of  this  diagonal  may  be  replaced 
by  any  quantities  whatever. 
By  what  precedes, 


a'  .,.  a 


(«) 


I  0  . 

. .  0 

Qa:. 

. .  ^,(«) 

Ta:. 

.  .  aJi"^^ 

m 


DETERMINANTS. 


[Chap.  IL 


Hence,  a  determinant  of  the  nth.  order  may  be  expressed 
as  a  determinant  of  the  order  (« -J"  i)  by  bordering  it  above 
by  a  row  (to  the  left  by  a  column)  of  zeros,  to  the  left  by  a 
column  (above  by  a  row)  of  elements  chosen  arbitrarily,  and 
writing  i  at  the  intersection  of  the  lines  thus  added.  By  con- 
tinuing this  process  any  determinant  may  be  expressed  as  a 
determinant  of  any  higher  order. 

Prob.  26.  If  all  the  elements  on  one  side  of  the  secondary  diag- 
onal are  zeros,  what  is  the  value  of  the  determinant  ? 

Prob.  27.  Develop  the  determinant 


a 

h 

g 

u 

0 

h 

b 

f 

V 

0 

g 

f 

c 

w 

0 

u 

V 

w 

0 

/ 

0 

0 

0 

/ 

s 

Prob.  28.  A  determinant  in  which  ak'''^  =  —  Us^^^  and   a^ 


(*) 


=  o  is 


said  to  be  skew-symmetric.     Prove  that  every  skew-symmetric  deter- 
minant of  odd  order  is  equal  to  zero. 

Art.  23.    Solution  of  Linear  Equations. 

Of  the  many  analytical  processes  giving  rise  to  determinants 
the  simplest  and  most  common  is  the  solution  of  systems  of 
simultaneous  linear  equations.     Thus,  solving  the  equations 

alx'  +  a;'x"  =  /c,,  * 
a,'x'  -f  a,''x"  =  AT,,  j 

by  the  methods  of  ordinary  algebra  gives  : 


In  the  notation  of  determinants  these  arc  written 


X'  — 


K,  a^ 


/f.  a^ 


I 


a'  a„ 


a'  K, 


/ 


It  will  be  noted  that  the  two  fractions  expressing  the  values 
of  x'  and  x"  have  a  common  denominator,  this  being  the  de- 
terminant whose  elements  are  the  coefficients  of  the  unknowns 
arranged  in  the  same  order  as  in  the  given  equations.     The 


Art.  23.] 


SOLUTION    OF    LINEAR    EQUATIONS. 


57 


numerator  of  the  fraction  giving  the  value  of  x'  is  formed  from 
this  denominator  by  replacing  each  coefficient  of  x'  by  the 
corresponding  absolute  term.     Similarly  for  x" , 

The  difficulty  of  solving  systems  of  linear  equations  by  the 
ordinary  processes  of  elimination  increases  rapidly  as  the  num- 
ber of  equations  is  increased.  The  law  of  formation  of  the 
roots  explained  above  is,  however,  capable  of  generaHzation, 
being  equally  applicable  to  all  complete  linear  systems,  as  will 
now  be  shown. 

Let  such  a  system  be  written 

a^x'  +  a^'x"  +  . .  .  +  ^/'";r(«>  =  at,  , 

/Co, 


a^x'  +  a^'x"  + 


+  ai^^x^^^ 

+  ajr'x^'"'  =  /c«. 


(I) 


Now  form  the  determinant  of    the  coefficients  of    these 
equations  ;  thus : 


n 


.  a 


a:  a. 


(«) 


(«) 


a^^Un    .  .  .a 


(«) 


and  let  A^'^  be  the  co-factor  of  ai'^  in  this  determinant.  The 
function 

a^(  p)A^^'^  +  ^,(^)  ^,w  +  .  .  .  +  a^^ ^,(*> 

is  equal  to  D  when/  =  s  (Art.  i8)  ;  to  zero  when  /  and  s  are 
different  superscripts  (Art.  19).  Then,  multiplying  the  given 
equations  by  A}'\  A^'\  .  .  .  A^'^  respectively,  the  sum  of  the 
resulting  equations  is  a  linear  equation  in  which  the  coefficient 
of  x'^'^  is  equal  to  D,  while  those  of  all  the  other  unknowns 
vanish.     The  sum  is,  therefore, 

nx^'^  =  K,A,^^^  +  K,A,^^^  +  .  .  .  +  ^„AJ^\  (2) 

But  the  second  member  of  this  equation  is  what  D  becomes 
upon  replacing  the  coefficients  a^^'\  a^'\  .  .  .aj^'^  of  the  unknown 
jir^*-  by  the  absolute  terms  /c, ,  /c, ,  .  .  .  /c„  in  order.     Hence, 


58 


DETERMINANTS. 


[Chap.  II. 


As)  _ 


a/  .  .  .  a,^'-'^K,a}'+'^  .  .  .  a. 


(«) 


(5-1) 


K„a. 


(^+1) 


,  .  .  a 


(«) 


(«) 


a,a^ 


< 


in) 


.  a 


(«) 


(3> 


This  result  may  be  stated  as  follows  : 

{a)  The  common  denominator  of  the  fractions  expressing 
the  values  of  the  unknowns  in  a  system  of  n  linear  equations 
involving  n  unknown  quantities  is  the  determinant  of  the 
coefficients,  these  being  written  in  the  same  order  as  in  the 
given  equations,  ifi)  The  numerator  of  the  fraction  giving  the 
value  of  any  one  of  the  unknowns  is  a  determinant,  which  may 
be  formed  from  the  determinant  of  the  coefificients  by  sub- 
stituting for  the  column  made  up  of  the  coefficients  of  the 
unknown  in  question  a  column  whose  elements  are  the  absolute 
terms  of  the  equations  taken  in  the  same  order  as  the  coeffi- 
cients which  they  displace. 

Prob.  29.  Solve  the  following  systems  of  equations  : 


(l) 

3*  +  5J  =  2i.     6x  +  2^=i5; 

(2) 

^f-.  7+-'^ 

(3) 

3^+:i;+25  =  50,     ^  + 2j;- 32=  15,'    2x-\-2y-  iz  =  2^\. 

(4) 

-;+7=A  r+j  =  ..  j+;-; 

(5) 

f+f+f+r"-  T+f+^^=- 

7  +  ^S  +  f3  =  -.  7  +  i^+rf  +  f,  =  - 

Prob.  30.  Show  that  the  three  right  lines 
_y  =  :x:+i,     y—  —  2X-\-\(i,     y 
intersect  in  a  common  point. 


3'^-  9. 


Art.  24.    Consistence  of  Linear  Systems. 

When  the  number  of  given  equations  is  greater  than  the 
number  of  unknowns  their  consistency  with  one  another  must 


Art.  24.] 


CONSISTENCE    OF    LINEAR    SYSTEMS, 


59" 


obviously  depend  upon  some  relation  among  the  coefficients. 
This  relation  will  now  be  investigated  for  the  case  of  {n  +  i) 
linear  equations  involving  n  unknowns.     Let  the  equations  be 


+  . . .  +  ^« 


(«)  ^(«) 


+  <\^^''^  = 


UnX 

If  the  above  equations  are  consistent  the  values  of  the  unknowns 
obtained  by  solving  any  n  of  them  must  satisfy  the  remaining 
equation.  Solving  the  first  n  equations  by  the  method  of  the 
preceding  article,  substituting  the  values  of  x' ^  x'\  . . .  jr^")  thus 
obtained  in  the  last  equation,  and  clearing  of  fractions,  the 
result  reduces  to  (Art.  i8) 

E~.  a!     ...  ^/"^    /c,       =  o, 


. .  a. 


(«). 


■«+i 


which  is  the  condition  to  be  fulfilled  by  the  coefficients  in  order 
that  the  given  equations  may  be  consistent. 

Hence  the  condition  of  consistency  for  a  set  of  linear  equa- 
tions involving  a  number  of  unknowns  one  less  than  the  number 
of  equations  is  that  the  determinant  of  the  coefficients  and 
absolute  terms,  written  in  the  same  order  as  in  the  given  equa- 
tions, shall  be  zero.  This  determinant  is  called  the  resultant* 
or  eliminant  of  the  equations.     Thus  the  equations 

x-\-y  —  z  =  o,  x— y-\-z=2,  —  x  -\-y  -\-z  =  4,  x  -\-y  -\-js  =  6 

are  consistent,  for  the  reason  that 

=  o. 


I 

I 

—  I 

0 

I 

—  I 

I 

2 

I 

I 

I 

4 

I 

I 

I 

6 

*  This  term  was  introduced  by  Laplace  in  1772.      The  term  eliminant  is 
due  to  Sylvester. 


60 


DETERMINANTS. 


[Chap.  II. 


Art.  25.    The  Matrix. 

Assume  r  linear  equations  involving  n  unknowns,  r  being 
greater  than  n.  as  follows : 


^,V  +  ...+^.(«>;r(«)  =  /r„,   y 


The  consistency  of  these  equations  requires  that  every  deter- 
minant of  the  order  («  +  0'  formed  by  selecting  {7i-{-  i)  rows 
from  the  array  whose  elements  are  the  coefficients  and  abso- 
lute terms  written  in  order,  shall  be  zero. 

If  the  elements  of  the  array  fulfill  this  condition  the  fact  is 
expressed  thus : 


^. 


aj    ...^/  I  =o; 


^/''>...^„<"\..^/"^ 


K,      ...  K^ 


.  ./f. 


the  change  of  rows  into  columns  being  purely  arbitrary.     The 
above  expression  is  called  a  rectangular  array,  or  a  matrix. 


Art.  26.     Homogeneous  Linear  Systems. 

Let  the  equations  of  the  given  system  be  both  linear  and 
homogeneous  ;  thus  : 

^/  ;tr'  +  .  .  .  +  a,^-^x^-^  =  O,  1 

h  (0 

Representing  the  determinant  of  the  coefficients  by  E,  the 
general  solution,  as  given  by  the  formula  (3)  of  Art.  23,  is 

,r(^)  =  o/£. 

That  is,  all  the  unknowns  are  equal  to  zero,  and  the  equa- 
tions have  no  other  solution  than  this  unless 

E  =  o, 


Art.  26.] 


HOMOGENEOUS    LINEAR    SYSTEMS. 


61 


But  in  this  case  the  value  of  each  unknown  is  obtainable 
only  in  the  indeterminate  form  o/o.  The  ratios  of  the  un- 
knowns may  be  readily  obtained,  however.  For,  dividing  each 
equation  through  by  any  one  of  these,  as  x^'\  the  system  (i) 
becomes 


As-1) 


+<-'z^+^- 


(S+1) 


;^(.+  l) 


+  ^. 


(«)■ 


(s) 


/  ^ 


(S-l) 


As+1) 


«.>+---+'^»"-';:^ +''«"■ 


is) 


+ 


.(«) 


+^'n'^,  =  -^ 


(s)       I 

J 


[    (2> 

I 


X'''  X 

Now  the  condition  £  =  o  establishes  the  consistency  of  the 
n  equations  (2)  involving  the  {n  —  i)  unknown  ratios  (Art.  24), 

Hence,  if  £  =  o  the  given  equations  (l)  are  consistent ;  that  is, 
the  values  of  the  above  {n  —  i)  ratios  obtained  by  solving  any 
(n  —  i)  of  them  will  satisfy  the  remaining  equation.  Any  n 
quantities  having  among  themselves  the  ratios  thus  determined 
will  satisfy  the  given  equations.  Thus,  if  xjj  xj\  .  .  .  x^^""^  are 
n  such  quantities,  so  also  are  XxJ,  A  x/' ,  .  .  .  \xj''*\  A  being  any 
factor  whatever. 

The  determinant  E  of  the  coef!icients  of  the  given  homo- 
geneous linear  equations  is  called  the  resultant  or  eliminant  of 
the  system. 

When  the  number  of  equations  is  greater  than  the  number 
of  unknowns  the  conditions  of  consistency  are  expressible  in 
the  form  of  a  rectangular  array,  as  in  Art.  25. 

As  an  example,  consider  the  five  equations 

2x  —  :^y  +  2  =  0,     4x  —  J/  —  2  =  o,     —  7^  +  3y  +  ^  =  Oy 
X  -{-  y  —  z  =  o,      ^x  —  ^y  -\-  z  =  o. 
Dividing  each  of  the  first  two  equations  by  z  and  solving 


for  the  two  unknowns  -  and  -  gives 

z           z 

X 

-1-3 

/ 

2-3 

_2      ^_ 

2  -  I 

/ 

2  - 

-3 

_3 

z 

I  —  I 

4-  I 

5'    ^ 

4       I 

4- 

-  I 

5 

82 


DETERMINANTS. 


[Chap.  H. 


or  ;tr:j:<3:::2:  3  :  5; 

and  any  three  quantities  having  these  ratios  will  satisfy  the 

two  equations,  as  lo,  15,  and  25.     That  the  third  equation  is 

consistent  with  the  first  two  is  shown  by  the  vanishing  of  the 

determinant 

2  —  3        I  I  =  o. 

4-  I  -  I 

-7       3       I 

If  all  the  equations  are  consistent  the  determinant  of  the 
coefficients  of  any  three  of  them  must  vanish ;  that  is, 


2       4-7 
-3-1       3 


I       5 
I  -5 


I  —  I 


o. 


Art.  27.    Co-factors  in  a  Zero  Determinant. 

If,  in  the  preceding  article,  ^  =  o,  it  follows  from  Arts.  18 
and  19  that 

a,' A/  +  a/'A/'  +  .  .  .  +  a.^^'W'  =  o, 

^/^/  +  a/' A,''  +  .  .  .  +  ^.^"^^/"^  =  o=£, 

aJA^  +  aJ'A,"  +  •  •  •  +  ^Ji'"'^^'''  =  O. 
These  equations  obviously  give  for  the  ratios 

Al_  A_t^     ^(^)  ^) 

values  which  are  identical  with  those  obtained  for  the  ratios 

\ 

from  the  equations  (i)  of  Art.  26.  It  follows  that  x\  x", . .  .  x^""^ 
are  proportional  to  ^/,  ^/',  .  .  .  A^''\  whatever  the  value  of  k. 
Thus,  giving  to  k  the  successive  values  i,  2,  .  .  .  «,  there  result 

x'  :x"  :.  .  ,  :x^^^'.'.A,':A^':.  .  .  r^/") 

:'.A::A:'-.,.:A^-^ 

ixAJ'.AJ'-..  ..  '.A^^\ 


Art.  28.  J 


Sylvester's  method  of  elimination. 


63 


Hence,  when  a  determinant  is  equal  to  zero,  the  co-factors 
of  the  elements  of  any  line  are  proportional  to  the  co-factors  of 
the  corresponding  elements  of  any  parallel  line.  r~ 


Art.  28.    Sylvester's  Method  of  Elimination.* 

Let  it  be  required  to  eliminate  the  unknown  from  the  two 

equations 

a^x^  +  a^x""  -\-  a,x  -{-  a^  =  o, 

d,x'  +  d,x  +  d,=:0. 

This  will  be  done  by  what  is  called  the  dialytic  method,  the 
invention  of  which  is  due  to  Sylvester.     Multiplying  the  first 
of  the  given  equations  by  ;r,  and  the  second  by  x  and  x^  suc- 
•cessively,  the  result  is  a  system  of  five  equations,  viz.: 
a^x'  -\-  a^x"^  +  a^x  -|-  ^„  =  o,  -^ 
a^x*  +  a^x^  -\-  a^x'  +  a,x  =  o, 

by  -h  b,x  -\-b,  =0, 
by  +  by  +  b,x  =  o, 

by  +  by  +  by  =  o.  ^ 

The   eliminant   of    these   five   equations,    involving   the    four 
•unknowns  ;r,  x\  x\  and  x*  is  (Art.  24) 

£  =  o  a^  a^  a^  a^    =  o. 

^3  a^  a,  a,  o 

o  o  b^  b,  b^ 

o  b^  b^  b,  o 

b^  b,  b,  o    o 

'If  the  given  equations  be  not  consistent  this  determinant  will 
not  vanish. 

The  above  method  is  a  general  one.     Thus,  let  the  two 
given  equations  be 

a^x"^  +  . , ,  +  a,x  +  a,  =  Oy 

bnx""  + ^b,x-Yb,  =  o. 

Multiplying  the  first  equation  (n  —  i)  times  in  succession 
iby  X,  and  the    second  (m  —  i)  times,  {m  -f  ti)    equations   are 
*  PhJinpophical  Magazine,  1840,  and  Crelle's  Journal,  Vol.  XXI. 


64  DETERMINANTS.  [ChAP.  IL 

obtained  which  involve  as  unknowns  the  first  {m  -\-  n  —  i) 
powers  ot  x.  The  eliminant  of  these  equations  is  a  determinant 
of  the  order  {m  -f-  n),  which  is  of  the  nth  degree  in  terms  of 
the  coefficients  of  the  equation  of  the  mth  degree,  and  vice 
versa.     The  law  of  formation  of  the  ehminant  is  obvious. 

The  same  method  may  be  used  in. eliminating  one  or  both 
the  variables  from  a  pair  of  homogeneous  equations. 

As  an  example,  let  it  be  required  to  eliminate  the  variables 
from  the  equations 

2x^  —  i,xy  —  9j^^  =  o     and     ^x^  —  Jxy  —  6y  =  o. 

Dividing  the  first  byj/',  and  multiplying  by  — ;  the  second 

y 

X 

by  y,  and  multiplying  by  —  twice  in  succession,  there  result, 


X    X      X  X 

in  all,  five  equations  involving  — ,  — ,  -v»  and  -j- 
these  four  ratios  gives 


EUminating 


E^ 


5  o  -  9 
0  —  9  o 
3-7-6 
7  —  60 


3-7-6 


o 


the  vanishing  of  which  shows  that  the  two  given  equations  are 
consistent. 

Prob.  31.  Test  the  consistency  of  each  of  the  following  systems 
of  equations: 

(i)  x-\-y  -\-  2Z  =g,  X  -\-y  —  z  —o^  2X  —  y  -{-  z  =^,  x—^-\-2Z=j;. 

(2)  X  —y  —  2Z  =0,       x  —  2y  -\r  z  =^  o,       2x  —  ^y  —  z  =  o; 

(3)  2xy  -  xf  =  o,       Sx'y  +  8^/  -  5/  =  o. 

Prob.  32.  Find  the  ratios  of  the  unknowns  in  the  equations 
2X  -{-y  —■  2Z  =:  Oy      ^7V  —  y  —  ^Z  =^  O,      2W  -\-  X  —  t^v  ^  z  ^=^  o. 

Prob.  33.  In  the  equations 

ak'x'  +  .  .  .  ^^^^"^^^"^  +  ^^^^  +^)x(«+^^  =0,     [/I  =  I,  2,  .  .  .  n'\ 
prove    that    jc'  :  .  .  .  :  x^''^  :  ^(''+'> ::  M'  :  .  .  .  :  M^""^ :  J/(''+^) ,   where 


Art.  29.] 


THE    MULTIPLICATION    THEOREM. 


66 


(— i)'-'J/^*'^  is  the  determinant  obtained  by  deleting  the  /th  column 
from  the  rectangular  array 


(«)^j^+i) 


an  ...  an    at 
lx-\-yy  -f  yus  _rx  -\-  jny  -\-\z  _  fAx-\-\y  +  nz 


Prob.  34.  From 
deduce 


Prob.  35.  Show  that  the  three  straight  lines  a'x  +  b'y  +  /  =  o, 
a"x  +  b"y  +  ^"  =  o,    and  «'"x  +  b'"y  -\-  c'"  =  o,  are  concurrent 
when  I  a'b"c"'  \  =  o. 

Prob.  36.  Prove  that  the  medians  of  a  triangle  are  concurrent. 

Prob.  37.   Show  that  the  points  (^o>^o)>  (•^if>'i)»  ^^^  i^^fy-t)   are 


X 

y 

z 

y  p,p 

I  ^p 

I  y  p 

tnX  q 

y  \  q 

y  m  q 

\  n  r 

ixnr 

//  A  r 

collinear  when 


o. 


x,y,  I 
^xy,  I 

x^y^  I 

Prob.  38.  Write  the  conditions  that  all  the  points  {x^^y^^ 
{^i^y^y  '  '  •  (^ntyn)  shall  bc  colUncar  in  the  form  of  a  matrix. 

Prob.  39.  Obtain  the  equation  of  a  right  line  through  (x^^y^} 
and  (x^,y^)  in  the  form  of  a  determinant. 


Prob.  40.  Show  that  the  equation 


X  y  z  I 
X,  y,  z^  I 
x^  y,  z^  I 


represents  a  plane  through  {x.^y^j  z^  (x,,y^,  ^J,  and  (^3,  j/,,  z^). 

Art.  29.    The  Multiplication  Theorem. 
Let  the  two  homogeneous  linear  equations 

a,,x,  +  a,,x,  =  o,  J 
be  subjected  to  linear  transformation  by  substituting 


(I) 


(2) 


66  DETERMINANTS. 

The  result  of  such  transformation  is 


[Chap.  II. 


(3) 
(4) 


{a  J,,  +  aj,,)u,  +  {a,,d,,  +  ajjii,  =  o, 
(^,/n  +  ^Ji,)m,  +  {a,A^  +  aj,,)zc,  =  o. 
The  vanishing  of  the  determinant 

is  the  condition  that  the  equations  (3)  may  be  consistent ;  that 
is,  the  condition  that  they  may  have  solutions  other  than 
u^  =z  o  =z  u^  (Art.  26).  Now  the  equations  (3)  may  be  consist- 
ent because  of  the  consi.stency  of  the  equations  (i),  in  which 
case  the  determinant 

vanishes.  Or,  this  condition  failing,  and  the  equations  (i) 
thus  having  no  solution  other  than  x^  =  o  =  x^,  the  equations 
(3)  will  still  be  consistent  if  the  equations  (2)  are  so ;  that  is, 
if  the  determinant 


b„  b^ 


(6) 


vanishes.  The  vanishing  of  either  of  the  determinants  (5)  or 
(6),  therefore,  causes  the  determinant  (4)  to  vanish.  It  follows 
that  (5)  and  (6)  are  factors  of  (4) ;  and  since  their  product  and 
the  determinant  (4)  are  of  the  same  degree  with  respect  to 
the  coefficients  ^„ ,  .  .  . ,  ^„ ,  .  .  . ,  they  are  the  only  factors. 
Hence, 


^n^„ 

. 

b.A, 

= 

^,,^„ 

b,A, 

(7) 


The  above  method  is  equally  applicable  to  the  formation 
of  the  product  of  any  two  determinants  of  the  same  order. 
Hence  results  the  following  general  formula: 

1  ail  a-22  .  .  .  a„n  ]  '   |  ^n  -^aa  .  .  .  Kn  j   = 


aii6ii+  ...  +aindin      aiid2i+  .  .  .  +au>hn  •    .    .    .  Clubm-^  . . .  +ainbnn 


an\bii  +  .  . .  +annbin      «ni<^2i  +  •     .  +  ^wi/i 


«»i(5ni+  .  .  .  +annbn 


■    (8) 


Art.  29.] 


THE    MULTIPLICATION    THEOREM. 


67 


The  process  indicated  by  this  formula  may  be  described  as 
follows :  * 

To  form  the  determinant  |/,^„  | ,  which  is  the  product  of 
two  determinants  \a^^n\  and  |^,,„|,  first  connect  by  plus  signs 
the  elements  in  the  rows  of  both  |^j„|  and  |/^,„| .  Then  place 
the  first  row  of  \a^^n\  upon  each  row  of  |^,„|  in  turn  and  let 
each  two  elements  as  they  touch  become  products.  This  is 
the  first  row  of  |/,,„|.  Perform  the  same  operation  upon  |<^,„| 
with  the  second  row  of  | a^.^ \  to  obtain  the  second  row  of  |/j „ | ; 
and  again  with  the  third  row  of   |^j,„|to  obtain  the  third  row 

of|A.«|;  etc. 

Any  element  of  this  product  is 

A.  =  ^*i^M  +  ^*,^«  +  •    •   •   +  ^knbsn-  (9) 

When  the  two  determinants  to  be  multiplied  together  are 
of  different  orders  the  one  of  lower  order  should  be  expressed 
as  a  determinant  of  the  same  order  as  the  other  (Art.  22),  after 
which  the  above  rule  is  applicable. 

The  product  of  two  determinants  may  be  formed  by 
columns,  instead  of  by  rows  as  above.  In  this  case  the  result 
is  obtained  in  a  different  form.  Thus  the  product  of  the  de- 
terminants  (5)  and  (6)  by  columns  is 


Prob.  41.  Form  the  following  products 


(i) 


a  h  g 
h  b  f 
S  f  c 


a  g 

g  <: 


(2) 


b  f 
f  c 


^  g 

g  c 


a  h 
h  b 


13) 


^,1  <2;„  «„ 

^.1  ^sa  <35,3 


^11     ^\1    -^13 

^i\  ^11  ^^ii 

^Jl     ^32    ^33 


;      (4) 

a,  b,  c,  1 . 

Oil 

a.Kc.\ 

I     0     I 

«,  b^  C,  1 

I     I     0 

Prob.  42.  Generalize  the  last  example  (see  Prob.  22,  Art.  18). 
Prob.  43.  By  forming  the  product 


a-[-bV-  I 


c-\-dV-  I 
a-bV^^ 


*  Carr's  Synopsis  of  Pure  Mathematics,  London,  1886,  Article  570. 


68 


DETERMINANTS. 


[Chap.  II. 


show   that   the   product   of   two  numbers,   each  the  sum  of  four 
squares,  is  itself  the  sum  of  four  squares. 

Art.  30.    Product  of  Two  Arrays. 

The  process  explained  in  the  preceding  article  may  be  ap- 
plied to  form  what  is  conventionally  termed  the  product  of 
two  rectangular  arrays.  It  will  appear,  however,  that  multi- 
plying two  such  arrays  together  by  columns  leads  to  a  result 
radically  different  from  that  obtained  when  the  product  is 
formed  by  rows. 

Let  the  two  rectangular  arrays  be 


^n^ia'^^s 


and 


b^fi^jj^^ 


The  product  of  these  by  columns  is 

^l/l3    +    ^2/23  ^12<^,3    +    ^2/23  ^13^13    +    ^23^23 

The  determinant  ^  is  plainly  equal  to  zero,  being  the  prod- 
uct of  two  determinants  formed  by  adding  a  row  of  zeros  to  one 
of  the  given  rectangular  arrays  and  a  row  of  elements  chosen 
arbitrarily  to  the  other. 

In  general,  the  product  by  columns  of  two  rectangular 
arrays  having  m  rows  and  n  columns,  m  being  less  than  n,  is  a 
determinant  of  the  n^^  order  whose  value  is  zero. 

Multiplying  together  the  above  rectangular  arrays  by  rows, 
the  result  is 


/I' 


^n^n  +  ^12^^12  +  ^13<^13 

a  J..   +   ^22'^,2   +   ^23^13 


'^2.^21    +    ^22^^22   +    ^23<^2 


^12^13 


bj,, 
b„b„ 


+ 


^11^13 
<^21^2  3 


^21^22 


In  the  same  manner  it  ^Tiay  be  shown  that  the  product  by 
rows  of  two  rectangular  arrays  having  m  rows  and  n  columns, 
m  being  less  than  n,  is  a  determinant  of  the  m^^  order,  which 
may  be  expressed  as  the  sumof  the  n  \/m !  {it  —  m) !  determinants 


Art.  31.] 


RECIPROCAL    DETERMINANTS. 


69 


formed  from  one  of  the  arrays  by  deleting  {n  —  m)  columns, 
each  multiplied  by  the  determinant  formed  by  deleting  the 
same  columns  from  the  other  array. 

Art.  31.    Reciprocal  Determinants. 

The  determinant  formed  by  replacing  each  element  of  a 
given  determinant  by  its  co-factor  is  called  the  reciprocal  of 
the  given  determinant."^     Thus,  the  reciprocal  of 


6^ 


.  a. 


is 


Z/  = 


^n^n  •  •  •  ^i« 
-^ai-^aa  •  •   •  -^aw 


^Mi^wa   •   •   .  ^nn  -^«i^Ma  •   •   •  -^nn 

The  product  of  these  two  determinants  is 

«3i^ii+  .  .  .  +aan-^in     ^21^21+  .  .  .-j-a^nAqn  ....  «ai-^ni4-.  •  '+a^n-^nn 
aniAii-{-.  .  .  .-\-annAin     ani^ai+.  .  .+«nn'^an.  •  •  •  an\An\+.  •  '-\-cinnAnn 

Each  element  on  the  principal  diagonal  of  this  product  is 
equal  to  S  (Art.  i8),  while  all  the  other  elements  vanish  (Art. 
19).     Hence, 


6, A  = 


6  o  .  .  .  o 
o   6  .  ,  .0 


,(«) 


=  d",     or     A  =  6' 


o„o  .  .  .  6 

That  is,  the  reciprocal  of  a  determinant  of  the  «*^  order  is 
equal  to  its  {n  —  i)*^  power. 

*  The  term  reciprocal  as  here  used  has  reference  to  the  algebraic  transforma- 
tion concerned  in  the  passage  from  point  coordinates  to  line  coordinates,  called 
reciprocation.  The  reciprocal  of  a  determinant  is  also  called  the  determinant 
adjugate. 


70  PROJECTIVE    GEOMETRY.  [ChAP.  III. 


Chapter   III. 

PROJECTIVE  GEOMETRY. 

By  George  Bruce  Halsted, 

Professor  of  Mathematics  in  the  University  of  Texas. 

Art.  1.    The  Elements  and  Primal  Forms  * 

1.  A  line  determined  by  two  points  on  it  is  called  a 
*  straight.* 

2.  On  any  two  points  can  be  put  one,  but  only  one,  straight, 
their  '  join.' 

3.  A  surface  determined  by  three  non-costraight  points  on 
it  is  called  a  *  plane.* 

4.  Any  three  points,  not  costraight,  lie  all  on  one  and  only 
one  plane,  their  'junction.' 

5.  If  two  points  lie  on  a  plane,  so  does  their  join. 

6.  The  plane,  the  straight,  and  the  point  are  the  elements 
in  projective  geometry. 

7.  A  straight  is  not  to  be  considered  as  an  aggregate  of 
points.  It  is  a  monad,  an  atom,  a  simple  positional  concept  as 
primal  as  the  point.  It  is  the  '  bearer  '  of  any  points  on  it.  It 
is  the  bearer  of  any  planes  on  it. 

8.  Just  so  the  plane  is  an  element  coeval  with  the  point.  It 
is  the  bearer  of  any  points  on  it,  or  any  straights  on  it. 

9.  A  point  is  the  bearer  of  any  straight  on  it  or  any  plane 
on  it. 

10.  A  point  which  is  on  each  of  two  straights  is  called 
their  '  cross.' 

*  This  Chapter  treats  Projective  Geometry  entirely  by  the  synthetic  method. 
Metric  relations  are  not  considered,  and  nothing  is  borrowed  from  Analytic 
Geometry. 


Art.  1.]  THE    ELEMENTS    AND    PRIMAL    FORMS.  71 

11.  Planes  all  on  the  same  point,  or  straights  all  with  the 
same  cross,  are  called  *  copunctal/ 

12.  Any  two  planes  lie  both  on  one  and  only  one  straight, 
their  *  meet.' 

13.  Like  points  with  the  same  join,  planes  with  the  same 
meet  are  called  costraight. 

14.  A  plane  and  a  straight  not  on  it  have  one  and  only  one 
point  in  common,  their  *  pass.' 

15.  Any  three  planes  not  costraight  are  copunctal  on  one 
and  only  one  point,  their  '  apex.' 

16.  While  these  elements,  namely,  the  plane,  the  straight, 
and  the  point,  retain  their  atomic  character,  they  can  be  united 
into  compound  figures,  of  which  the  primal  class  consists  of 
three  forms,  the  *  range,'  the  *  fiat-pencil,'  the  *  axial-pencil.' 

17.  The  aggregate  of  all  points  on  a  straight  is  called  a 

*  point-row,'  or  '  range.'     If  a  point  be  common  to  two  ranges, 
it  is  called  their  *  intersection.' 

18.  A  piece  of  a  range  bounded  by  two  points  is  called  a 
'  sect.' 

19.  The  aggregate  of  all  coplanar,  copunctal  straights  is 
called  a  *  flat-pencil.'  The  common  cross  is  called  the  '  pencil- 
point.*     The  common  plane  is  Called  the  *  pencil-plane.' 

20.  A  piece  of  a  flat-pencil  bounded  by  two  of  the  straights, 
as  *  sides,'  is  called  an  *  angle.* 

21.  The  aggregate  of  all  planes  on  a  straight  is  called  an 

*  axial-pencil,*  or  'axial.*     Their  common   meet,  the  *  axis,*  is 
their  bearer. 

22.  A  piece  of  the  axial  bounded  by  two  of  its  planes,  as 
sides,  is  called  an  *  axial  angle.' 

23.  Angles  are  always  pieces  of  the  figure,  not  rotations. 

24.  No  use  is  made  of  motion.  If  a  moving  point  is  spoken 
of,  it  is  to  be  interpreted  as  the  mind  shifting  its  attention. 

25.  When  there  can  be  no  ambiguity  of  meaning,  a  figure 
in  a  pencil,  though  consisting  only  of  some  single  elements  of 
the  complete  pencil,  may  yet  itself  be  called  a  pencil.  Just  so, 
certain  separate  costraight  points  may  be  called  a  range. 


72  PROJECTIVE    GEOMETRY.  [ChAP.  III. 

Art.  2.     Projecting  and  Cutting. 

26'.  To  *  project  *  from  a  fixed  point  M  (the  *  projection- 
vertex  ')  a  figure,  the  *  original,'  composed  of  points  B,  Cy  Z>, 
etc.,  and  straights  b,  c,  d,  etc.,  is  to  construct  the  '  projecting 
straights  '  MB,  ~MC,  MD,  and  the  *  projecting  planes '  Mb,  Mc, 
Md,  Thus  is  obtained  a  new  figure  composed  of  straights  and 
planes,  all  on  M^  and  called  an  '  eject '  of  the  original. 

27.  To  'cut'  by  a  fixed  plane  //  (the  '  picture-plane ')  a 
figure,  the  *  subject,'  made  up  of  planes  /?,  y,  ^,  etc.,  and 
straights  b,  c,  d,  etc.,  is  to  construct  the  meets  /^^,  ///,  yud',  and 

the  passes  }xb,  jxcy  fxd.  Thus  is  obtained  a  new  figure  com- 
posed of  straights  and  points,  all  on  /f,  and  called  a  '  cut '  of 
the  subject.  If  the  subject  is  an  eject  of  an  original,  the  cut 
of  the  subject  is  an  *■  image '  of  the  original. 

28.  Axial  projection.  To  project  from  a  fixed  straight  m 
(the  '  projection-axis  '),  an  original  composed  of  points  B^  C,  Z>, 
etc.,  is  to  construct  the  projecting  planes  inB,  mC,  mD.  Thus 
is  obtained  a  new  figure  composed  of  planes  all  on  the  axis  m, 
and  called  an  *  axial-eject '  of  the  original. 

29.  To  cut  by  a  fixed  straight  m  (to  *  transfix  ')  a  subject 
composed  of  planes  y5,  y,  S^  etc.,   is  to  construct  the  passes 

w/?,  my,  md.  The  cut  obtained  by  transfixion  is  a  range  on 
the  *  transversal  *  m. 

30.  Any  two  fixed  primal  figures  are  called  *  projective  ' 
(a")  when  one  can  be  derived  from  the  other  by  any  number  of 
projectings  and  cuttings.* 

Art.  3.    Elements  at  Infinity. 

31.  It  is  assumed  that  for  every  element  in  either  of  the 
three  primal  figures  there  is  always  an  element  in  each  of  the 
others. 

*Pascal  (1625-62)  and  Desargues  (1593-1662)  seem  to  have  been  the  first  to 
derive  properties  of  conies  from  the  properties  of  the  circle  by  considering  the 
fact  that  these  curves  lie  in  perspective  on  the  surface  of  the  cone. 


Art.  3.]  ELEMENTS    AT    INFINITY.  73 

32.  On  each  straight  is  one  and  only  one  point  *  at  infinity,' 
or  '  figurative'  point.  The  others  are  '  proper  '  points.  Any 
point  going  either  way  (moving  in  either  '  sense ')  ever  forward 
on  a  straight  is  at  the  same  time  approaching  and  receding 
from  its  point  at  infinity.  The  straight  is  thus  a  closed  line 
compendent  through  its  point  at  infinity. 

33.  *  Parallels '  are  straights  on  a  common  point  at  infinity. 

34.  Two  proper  points  in  it  divide  a  range  into  a  finite  sect 
and  a  sect  through  the  infinite.  Its  figurative  point  and  a 
proper  point  in  it  divide  a  range  into  two  sects  to  the  infinite 
<*  rays'). 

35.  All  the  straights  parallel  to  each  other  on  a  plane  are  on 
the  same  point  at  infinity,  and  so  form  a  flat-pencil  whose  pen- 
<:il-point  is  figurative.  Such  a  pencil  is  called  a  '  parallelflat- 
pencil.' 

36.  All  points  at  infinity  on  a  plane  lie  on  one  straight  at 
infinity  or  figurative  straight.*  Its  cross  with  any  proper 
straight  on  the  plane  is  the  point  at  infinity  on  the  proper 
straight. 

37.  Parallel-flat-pencils  on  the  same  plane  have  all  a 
straight  in  common,  namely,  the  straight  at  infinity  on  which 
are  the  figurative  pencil-points  of  all  these  pencils. 

38.  Two  planes  whose  meet  is  a  straight  at  infinity  are 
•called  parallel. 

39.  All  the  planes  parallel  to  each  other  are  on  the  same 
figurative  straight,  and  so  form  an  axial  pencil  whose  axis  is  at 
infinity.     Such  an  axial  is  called  a  parallel-axial. 

40.  All  points  at  infinity  and  all  straights  at  infinity  lie  on 
a  plane  at  infinity  or  figurative  plane.  This  plane  at  infinity  is 
common  to  all  parallel-axials,  since  it  is  on  the  axis  of  each. 

Prob.  I.  From  each  of  the  three  primal  figures  generate  the  other 
two  by  projecting  and  cutting. 

*  This  statement  should  not  be  interpreted  as  descriptive  of  the  nature  of 
infinity.  In  the  Function  Theory  it  is  expedient  to  consider  all  points  in  a 
plane  at  infinity  as  coincident. 


74  PROJECTIVE    GEOMETRY.  [Chap.  III. 

Art.  4.    Correlation  and  Duality. 

41.  Two  figures  are  called  *  correlated '  when  every  element 
of  each  is  paired  with  one  and  only  one  element  of  the  other. 
Correlation  is  a  one-to-one  correspondence  of  elements.  The 
paired  elements  are  called  *  mates.' 

42.  Two  figures  correlated  to  a  third  are  correlated  to  each 
other.  For  each  element  of  the  third  has  just  one  mate  in 
each  of  the  others,  and  these  two  are  thus  so  paired  as  to  be 
themselves  mates. 

43.  On  a  plane,  any  theorem  of  configuration  and  deter- 
mination, with  its  proof,  gives  also  a  like  theorem  with  its 
proof,  by  simply  interchanging  point  with  straight,  join  with 
cross,  sect  with  angle.* 

This  correlation  of  points  with  straights  on  a  plane  is 
termed  a  *  principle  of  duality.'  Each  of  two  figures  or  theo- 
rems so  related  is  called  the  '  dual '  of  the  other.f 

Prob.  2.  When  two  coplanar  ranges  m^  and  m'  are  correlated  as 
cuts  of  a  flat-pencil  J/,  show  that  the  figurative  point  T',,  or  Q' y  of 
the  one  is  mated,  in  general,  to  a  proper  point  P\  or  Q^ ,  of  the 
other. 

Prob.  3.  Give  the  duals  of  the  following: 

i'.  Two  coplanar  straights  determine  a  flat-pencil  on  their  cross. 
2'.  Two  coplanar  flat-pencils  determine  a  straight,  their  *  concur.^ 
3,.   Two  points  bound  two  '  explemental '  sects. 
Prob.  4.   To  draw  a  straight  crossing  three  given  straights,  join 
the  passes  of  two  with  a  plane  on  the  third.     Give  and  solve  dual. 

Art.  5.     PoLYSTiMS  and  Polygrams. 

44,.  A  *  polystim  '  is  a  system  of  44'.  A  *  polygram  '  is  a  system 

n  coplanar  points  (*  dots  '),  with  of  n  coplanar  straights  (*  sides  '), 

all    the    ranges    they    determine  with  all  the  flat-pencils  they  de- 

(*  connectors  ').    Assume  that  no  termine  (*  fans  ').      Assume  that 

three  dots  are  costraight.  no  three  sides  are  copunctal. 

*Culmann's  Graphic  Statics  (Zurich,  1864)  made  extensive  use  of  the 
theorems  of  duality.  Reye's  Geometric  der  Lage  (Hannover,  1866)  was  is- 
sued as  a  consequence  of  the  Graphic  Statics  of  Culmann. 

f  In  Algebra  the  principle  of  duality  consists  in  the  interpretation  of  the 
same  equation  in  different  kinds  of  coordinates — point  and  line  or  point  and 
plane  coordinates. 


Art.  5.] 


POLYSTIMS    AND    POLYGRAMS. 


75 


In  each  dot  intersect  («  —  i) 
connectors,  going  through  the  re- 
maining (n—  i)  dots.  So  there 
are  n{n  —  i)/2  connectors. 

45,.  For  n  greater  than  3,  the 
connectors  will  intersect  in 
points  other  than  the  dots.  Such 
intersections  are  called  '  codots.* 

46^.  There  are 
n{n  —  i)(«  —  2)(n  —  3)/8codots. 


In  each  side  concur  (n  —  i) 
fans,  going  through  the  remain- 
ing {n  —  i)  sides.  So  there  are 
n{n  —  i)/2  fans. 

45'.  For  n  greater  than  3,  the 
fans  will  concur  in  straights  other 
than  the  sides.  Such  concurs 
are  called  *  diagonals.' 

46'.  There  are 
n{n  —  i)(n  —  2)(«  —  3)/8  diago- 


nals. 
Proof  of  46,.  In  a  polystim  of  n  dots  there  are  n{n  —  i)/2 
connectors.     These  connectors  intersect  in 

[n{n  —  i)/2][n{n  —  i)/2  —  i]/2  =  «(;/  —  i)(«'  —  n—  2)/8 

points;  i.e.,  the  number  of  different  combinations  of  n{n  —  i)/2 
things,  two  at  a  time. 

But  some  of  these  intersections  are  dots,  and  the  remaining 
ones  are  codots.  Now  {n  —  i)  of  these  connectors  meet  at 
each  dot.  Therefore  each  dot  is  repeated  {n  —  i)  (n  —  2)j2 
times;  or  the  number  of  times  the  connectors  intersect  in 
points  not  codots,  i.e.  in  dots,  is  n{n  —  i)(«  —  2)/2. 

Therefore  the  number  of  codots  is 

n{n  —  \){n^  —  n  —  2)/8  —  n{n  —  i)(«  —  2)/2 
=  \n{n  —  i)/8][«'  —  n—  2  —  4^n  —  2)\ 
=  n{n  -i)(n-  2){n  —  3)/8. 


47,.  A  set  of  n  connectors  may 
be  selected  in  several  ways  so 
that  two  and  only  two  contain 
each  one  of  the  n  dots.  Such 
a  set  of  connectors  is  called  a 
*  complete  set '  of  connectors. 

48,.  There  are  («  —  i)  I/2 
complete  sets  of  connectors. 


47'.  A  set  of  n  fans  may  be 
selected  in  several  ways  so  that 
two  and  only  two  contain  each 
one  of  the  n  sides.  Such  a  set 
of  fans  is  called  a  *  complete  set ' 
of  fans. 

48'.  There  are  («  —  i)  !/2 
complete  sets  of  fans. 


Proof  of  48,.  In  a  polystim  of  n  dots  there  are  through  any 
single  dot  (ft  —  i)  connectors,  and  hence  {n  —  i){n  —  2)/2: 
pairs  of  connectors.     Consider  one  such  pair,  as  BC  and  BE. 


76  PROJECTIVE    GEOMETRY.  [ChAP,  III. 

The  number  of  different  sets  (each  o(  n  —  2  connectors) 
from  C  to  E  through  A,  B,  F,  G,  etc.  [there  being  {n  —  3) 
such  dots],  is  (n  —  3)  !,  i.e.  the  number  of  permutations  of 
(;2  _  3)  things.  Hence  the  number  of  complete  sets  of  con- 
nectors having  the  pair  BC  and  BE  is  (/^  —  3)  I  Therefore  the 
whole  number  of  complete  sets  of  connectors  is 

{n  -  i)in  -  2)[{n  -  3) !  ]/2  =  {n  -  i)  !/2. 

49,.  In  any  complete  set  of  49'-  ^^  ^"Y  complete  set  of 
connectors,  when  n  is  even,  the  fans,  when  n  is  even,  the  first  and 
first  and  the  {n/2  +  i)th.  are  the  (n/2-{-i)th  are  called  'op- 
called  '  opposite  '.  posite.' 

5O1.  A  '  tetrastim  '  is  a  system  50'.  A  *  tetragram  '  is  a  system 
of  four  dots  with  their  six  con-  of  four  straights  with  their  six 
nectors.  Each  pair  of  opposite  fans.  Each  pair  of  opposite  fans 
connectors  intersect  in  a  codot.  concur  in  a  diagonal.  These 
These  three  codots  determine  three  diagonals  determine  the 
the  '  codot-tristim  '  of  the  tetra-  '  diagonal- trigram  '  of  the  tetra- 
stim. gram. 

51.  Two  correlated  polystims  whose  paired  dots  and  co- 
dots  have  their  joins  copunctal  are  called  '  copolar.' 

52.  Two  correlated  polystims  whose  paired  connectors  in- 
tersect and  have  their  intersections  costraight  are  called 
*  coaxal.* 

53.  If  two  non-coplanar  tristims  be  copolar,  they  are  coaxal. 
For  since  AA'  crosses  BB\  therefore  AB  and  A^B'  intersect  on 
the  meet  of  the  planes  of  the  tristims. 

54.  If  two  non-coplanar  tristims  be  coaxal,  they  are  copolar. 
For  since  AB  intersects  A'B\  these  four  points  are  coplanar. 
The  three  planes  ABA'B',  ACA'C,  BCB'C  are  copunctal. 
Hence  so  are  their  meets  AA\  BB' ,  CO. 

55.  By  taking  the  angle  between  the  planes  evanescent,  is 
seen  that  coplanar  coaxal  tristims  are  copolar  ;  and  then  by 
reductio  ad  absurdum  that  coplanar  copolar  tristims  are  coaxal. 

56.  If  two  coplanar  polystims  are  copolar  and  coaxal  they 
.are  said  to  be  *  complete  plane  perspectives.'     Their  pole  and 


Art.  6.]  harmonic  elements.  77" 

axis  are  called  the  *  center  of  perspective*  and  the  *  axis  of 
perspective.' 

57.  If  two  coplanar  tristims  are  copolar  or  coaxal,  they  are 
complete  plane  perspectives. 

58.  If  two  coplanar  polystims  are  images  of  the  same  poly- 
stim  from  different  projection  vertices  F^ ,  F^ ,  they  are  com- 
plete plane  perspectives.  For  the  joins  of  pairs  of  correlated 
points  are  all  copunctal  (on  the  pass  of  the  straight  F,  F, 
with  the  picture  plane),  and  the  intersections  of  paired  con- 
nectors are  all  costraight  (on  the  meet  of  the  picture  plane 
and  the  plane  of  the  original). 

Prob.  5.  In  a  hexastim  there  are  15  connectors  and  45  codots. 
In  a  hexagram  there  are  15  fans  and  45  diagonals. 

Prob.  6.  If  the  vertices  of  three  coplanar  angles  are  costraight^ 
their  sides  make  three  tetragrams  whose  other  diagonals  are  copunc- 
tal by  threes  four  times.     [Prove  and  give  dual.] 

Prob.  7.  The  corresponding  sides  of  any  two  funiculars  of  a 
given  system  of  forces  cross  on  a  straight  parallel  to  the  join  of  the 
poles  of  the  two  funiculars. 

Art.  6.    Harmonic  Elements. 

59.  Fundamental  Theorem. — If  two  correlated  tetrastims 
lie  on  different  planes  whose  meet  is  on  no  one  of  the  eight 
dots,  and  if  five  connectors  of  the  one  intersect  their  mates, 
then  the  tetrastims  are  coaxal.  For  the  two  pairs  of  tristims 
fixed  by  the  five  pairs  of  intersecting  connectors  being  coaxal 
are  copolar.     Hence  the  sixth  pair  of  connectors  are  coplanar. 

60.  If  the  tetrastims  be  coplanar,  and  if  five  intersections  of 
pairs  of  correlated  connectors  are  costraight,  this  the  coplanar 
case  can  be  made  to  depend  upon  the  other  by  substituting 
for  one  of  the  tetrastims  its  image  on  a  second  plane  meeting 
the  first  on  the  bearer  of  the  five  intersections. 

61.  If  the  axis  m  is  a  figurative  straight,  the  theorem  reads  v 
If  of  two  correlated  tetrastims  five  pairs  of  mated  connectors 
are  parallel,  so  are  the  remaining  pair. 

62.  Four  costraight  points  are  called  '  harmonic  points  '  or 


78 


PROJECTIVE    GEOMETRY. 


[Chap.  III. 


a  *  harmonic  range/  if  the  first  and  third  are  codots  of  a  tetra- 
stim  while  the  other  two  are  on  the  connectors  through  the 
third  codot. 

6^.  By  three  costraight  points  and  their  order  the  fourth 
harmonic  point  is  uniquely  determined.   For  if  the  three  points 


in  order  are  A,  B,  C,  draw  any  two  straights  through  A,  and  a 
third  through  B  to  cross  these  at  K  and  M  respectively.  Join 
CK,  crossing  AM  at  N.  Join  CMy  crossing  AK  at  L.  Then  the 
join  ZiV  crosses  the  straight  ABC,  always  at  the  same  point  D, 
the  fourth  harmonic  to  ABC  separated  from  B. 

64.  In  projecting  from  a  point  not  coplanar  with  it  a 
tetrastim  defining  a  harmonic  range,  the  four  harmonic  points 
are  projected  by  four  coplanar  straights,  called  *  harmonic 
straights  '  or  a  *  harmonic  flat-pencil.* 

65.  The  four  planes  projecting  harmonic  points  from  an 
axis  not  coplanar  with  them  are  called  *  harmonic  planes/  or  a 
*  harmonic  axial-pencil.' 

66.  Projecting  or  cutting  a  harmonic  primal  figure  gives 
always  again  a  harmonic  primal  figure. 

6y.  By  three  elements  of  a  primal  figure,  given  which  is  the 
second,  the  fourth  harmonic  is  completely  determined. 

68.  Defining  harmonic  points  by  the  tetrastim  distinguishes 


Art.  6.] 


HARMONIC    ELEMENTS. 


79 


two  points  made  codots  from  the  other  two.  Yet  it  may  be 
shown  that  the  two  pairs  of  points  play  identically  the  same_ 
role. 

First,  from  the  definition  of  four  harmonic  points  each  sep- 
arated two  may  be  interchanged  without  the  points  ceasing  to 


be  harmonic  [or,  if  ABCD  is  a  harmonic  range,  so  is  also 
ADCB,  CBAD,  and  CDAB].  For  the  first  and  third  remain 
codots. 

Second,  to  prove  that  in  a  harmonic  range  the  two  pairs  of 
separated  points  may  be  interchanged  without  the  four  points 
ceasing  to  be  harmonic  [or,  if  ABCD  is  a  harmonic  range 
(and  therefore  ADCB,  CBAD,  and  CDAB),  then  also  is  BADC, 
DABC,  BCD  A,  and  DCBA^  :  Through  the  third  codot  O  draw 
the  joins  AO  and  CO.  These  determine  on  the  connectors 
NK,  KL,  LM,  and  MN  four  new  points,  5,  T,  U,  F,  respec- 
tively. The  tetrastim  KTOS  has  for  two  codots  A  and  C,  and 
has  a  connector  though  B ;  hence  its  remaining  connector  TS 
must  pass  though  D.  In  like  manner,  the  connector  UV  of 
the  tetrastim  J/F(9 6^  must  pass  through  Dy  and  a  connector 
of  each  of  the  tetrastims  LUOT and  VNSO  through  B.  There- 
fore B  and  D  are  codots  of  a  tetrastim  STUV  with  the  remain- 
ing connectors,  one  through  A,  one  through  C. 

69.  The  separated  points  A  and  C  are  called  *  conjugate 
points,*  as  also  are  B  and  D.  Either  two  are  said  to  be  *  har- 
monic conjugates '  with  respect  to  the  other  two. 

Prob.  8.  To  determine  the  join  of  a  given  point  M  with  the  in- 
accessible cross  X  of  two  given  straights  n  and  n'. 


80 


PROJECTIVE    GEOMETRY. 


[Chap.  Ill, 


Through  MdrsLW  any  two  straights  crossing  nslB  and  £',  and  ?/ 
at  D  and  D'.  Join  DB  and  D' B' ,  crossing  on  A.  Through  A  draw- 
any  third  straight  crossing 
11  at  B"  and  n'  at  D", 
Join  ^'Z)"  and  Z>'^", 
crossing  at  Z.  Then  ZJ/ 
is  the  join  required. 

Proof.  The  tetrastim 
XBMD  makes  AB'C'D' 
a  harmonic  range,  as 
XB'LD'  does  AB"C"D".  But  projecting  AB"C"D"  from  X, 
and  cutting  the  eject  by  AB' D'  gives  a  harmonic  range.  Therefore 
C,  C\  and  X  are  costraight.* 

Prob.  9.  Through  a  given  point  to  draw  with  the  straight-edge 
a  straight  parallel  to  two  given  parallels. 

Prob.  10.  To  determine  the  cross  of  a  given  straight  m  with  the 
inconstructible  join  x  of  two  given  points  N  and  W.  Join  any  two 
points  on  m  with  JV 
and  N\  giving  b 
and  b'  on  iV,  ^  and 
^'  on  N'.  Join  the 
crosses  ^-!^  and  d'b' 
by  ^.  On  a  take 
any  third  point  join- 
ing with  JV  in  b" 
and  with  A"'  in  d'\ 

Join  the  crosses  <^V"  and  ^'^"  by  /. 
quired.     [From.  Prob.  8,  by  duality.] 

Prob.  II.  Cut  four  coplanar  non-copunctal  straights  in  a  har- 
monic range. 

Prob.  12.  On  a  given  straight  determine  a  point  from  which  the 
ejects  of  three  given  points  form  with  the  given  straight  a  harmonic 
pencil. 

Art.  7.  Projectivity. 
70.  Two  primal  figures  of  three  elements  are  always  pro- 
jective.— If  one  be  a  pencil,  take  its  cut  by  a  transversal.  If 
the  bearers  of  ABC  and  A'B'C  be  not  coplanar,  join  AA\ 
BB\  CC\  and  cut  these  joins  by  a  transversal,  m.  Then  ABC 
and  A'B'C  are  two  cuts  of  the  axial  mAA',  mBB' ,  mCC . 

*  Numerous  problems  in  Surveying  may  be  solved  by  the  application  of  the 
preceding  principles,  but  such  application  has  not  been  found  advantageous  in 
practice.     See  Gillespie's  Treatise  on  Land  Surveying,  New  York,  1872. 


Then   Im  is  the  cross  re- 


Art.  7.]  projectivity.  81 

If  the  bearers  are  coplanar,  take  on  the  join  AA^  any  two 
projection  vertices  M  and  M\  Join  MB  and  M' B' ,  crossing 
at  B"\  join  MC  and  M' C\  crossing  at  C" .  Join  B" C"  crossing 
AA'  at  A".     Then  ABC  and  A' B' C  are  images  of  A" B" C" . 

71.  If  any  four  harmonic  elements  are  taken  in  one  of  two 
projective  figures,  the  four  elements  correlated  to  these  are  also 
harmonic.  For  both  ejects  and  cuts  of  harmonic  figures  are 
themselves  harmonic. 

72.  Two  primal  figures  are  projective  if  they  are  so  corre- 
lated that  to  every  four  harmonic  elements  of  the  one  are 
correlated  always  four  harmonic  elements  of  the  other.  For 
the  same  projectings  and  cuttings  which  derive  A' B' C  from 
ABC^\\\  give  D^  from  D,  Therefore  A'B'C'D^  is  harmonic. 
But  by  hypothesis  A' B' C D'  is  harmonic.     Therefore  D^  is  D' , 

73.  If  two  primal  figures  are  projective,  then  to  every  con- 
secutive order  of  elements  of  the  one  on  a  bearer  corresponds 
a  consecutive  order  of  the  correlated  elements  of  the  other  on 
a  bearer. 

74.  Two  projective  primal  figures  having  three  elements 
self-correlated  are  identical.  For  two  self-correlated  elements 
cannot  bound  an  interval  containing  no  such  element,  since 
they  must  harmonically  separate  one  without  it  from  one 
within. 

75.  Two  ranges  are  called  '  perspective  '  if  cuts  of  the  same 
flat  pencil. 

Two  flat  pencils  are  perspective  if  cuts  of  the  same  axial 
pencil,  or  ejects  of  the  same  range.  Two  axials  are  perspective 
if  ejects  of  the  same  flat  pencil. 

A  range  and  a  flat  pencil,  a  range  and  an  axial  pencil,  or  a 
flat  pencil  and  an  axial  are  perspective  if  the  first  is  a  cut  of 
the  second. 

76,.  If  two  projective   ranges  76'.  If  two  coplanar  projective 

not  costraight  have  a  self-corre-  flat  pencils  not  copunctal    have 

lated  point  A^  they  are  perspec-  a  self-correlated  straight  a^  they 

tive.  are  perspective. 


82 


PROJECTIVR    GEOMETRY. 


[Chap.  III. 


Let  the  join  of  any  pair  of 
correlated  points  BB'  cross  the 
join  of  any  other  pair  CC  at  V. 

Projecting  the  two  given 
ranges  from  F,  their  ejects  are 
identical,  since  they  are  projec- 
tive and  have  the  three  straights 
VA,  VBB\  VCC  self-corre- 
lated. 


Let  the  cross  of  any  pair  of 
correlated  straights  bb'  join  the 
cross  of  any  other  pair  cc'  by  m» 

Cutting  the  two  given  flat  pen- 
cils by  w,  their  cuts  are  identical, 
since  they  are  projective  and 
have  the  three  points  ma^  mbb\ 
mcc'  self-correlated. 


Art.  8.    Curves  of  the  Second  Degree. 


77j.  If  two  coplanar  non- 
copunctal  flat  pencils  are  pro- 
jective but  not  perspective,  the 
crosses  of  correlated  straights 
form  a  *  range  of  the  second  de- 
gree,' or  *  conic  range. ' 


77'.  If  two  coplanar  non- 
costraight  ranges  are  projective 
but  not  perspective,  the  joins  of 
correlated  points  form  a  '  pencil 
of  the  second  class,'  or  *  conic 
pencil' 


78,.  If  two  copunctual  non- 
costraight  axial  pencils  are  pro- 
jective but  not  perspective,  the 
meets  of  correlated  planes  form 
a  *  conic  surface  of  the  second 
order,'  or  'cone.' 


78'.  If  two  copunctal  non- 
coplanar  flat  pencils  are  projec- 
tive but  not  perspective,  the 
planes  of  correlated  straights 
form  a  *  pencil  of  planes  of  the 
second  class,'  or  *  cone  of  planes.' 


79.  All  results  obtained  for  the  conic  range  or  the  conic 
pencil  are  interpretable  for  the  cone  or  cone  of  planes,  since 
the  eject  of  a  conic  is  a  cone  and  the  cut  of  a  cone  is  a  conic. 

80,.  On  the  cross  A  of  any  pair  80'.  On  the  join  a  of  any  pair 

of  correlated  straights  a  and   a^      of  correlated  points  A  and  A^  of 


Art.  8.] 


CURVES   OF    THE    SECOND    DEGREE. 


83 


of  the  projective  flat  pencils  V 


and    F,   draw   two    straights    u 
and  u^o 

The  cuts  ABC  and  A.B^C, 
being  projective  and  having  a 
pair  of  correlated  points  A,  A^ 
coincident,  are  perspective,  both 
being  cuts  of  the  pencil  on  F,, 
the  cross  of  the 
joins  BB^  and 

cc,. 

.  Any  straight 
^  of  F",  crossing 
u  at  jD,  is  then 
correlated  to 
the  join  of  F, 
with  the  cross 
Z>,  of  u^  and 
the  join  Z>F,. 
Any  ^  crosses 
its  ^,  so  deter- 
mined, at  P,  a  point  of  the  conic 
range  k. 

8 1,.  The  pencil-points  F,  Fi 
of  the  generating  pencils  pertain 
to  the  conic,  since  their  join 
W^  is  crossed  by  the  element 
correlated  to  it  in  either  pencil 
at  its  pencil-point. 


the  projective  ranges  u  and  u^ 
take  two  points  F  and  F^. 
The  ejects  adc  and  a^i^iC^ 
being  projective  and  hav- 
ing a  pair  of  correlated 
straights  a,  a^  coincident, 
are  perspective,  both  be- 
ing ejects  of  the  range  on 
2^2,  the  join  of  the  crosses 
bb^  and  cc^. 

Any  point  D  of  u^ 
joined  with  F  by  d^  is 
then  correlated  to  the 
cross  of  «i  with  the  join  d^  of  Fj 
and  the  cross  du^. 

Any  D  joined  to  its  Z>,  so  de- 
termined, gives  p  a  straight  of 
the  conic  pencil  K, 


8i'.  The  bearers  «,  «,  of  the 
generating  ranges  pertain  to  the 
conic,  since  their  cross  uu^  is 
joined  to  the  element  correlated 
to  it  in  either  range  by  its  bearer. 


84 


PROJECTIVE    GEOMETRY. 


[Chap.  Ill, 


82j.  The  straight  on  F  corre- 
lated to  V^  V  is  called  the  '  tan- 
gent' at  V.  Every  other  straight 
on  V  is  its  join  with  a  second 
point  of  the  conic. 

831.  On  any  straight,  as  «,  on 
any  point  A  of  the  conic,  its 
second  element  is  its  cross  M 
with  the  join  V^V^. 

84,.  From  the  five  given  points 
VV^  AML^  of  k  construct  a  sixth, 
F.  The  cross  D  oi  u  with  the 
join  VP^  and  the  cross  Z>,  of  u^ 
with  the  join  V^P  are  costraight 
with  Fj.  Therefore*  the  three 
opposite  pairs  in  every  complete 
set  of  connectors  of  a  hexastim 
whose  dots  are  in  a  conic  inter- 
sect in  three  costraight  codots 
whose  bearer  is  called  a  *  Pascal 
straight.* 

This  hexastim  has  sixty  Pascal 
straights,  since  it  has  sixty  com- 
plete sets  of  connectors. 

85^.  The  ejects  of  the  points 
of  a  conic  from  any  two  are  pro- 
jective. 

861.  By  five  of  its  points  a 
conic  is  completely  determined. 

87^.  Instead  of  five  points 
may  be  given  the  two  pencil- 
points  and  three  pairs  of  corre- 
lated straights.  If  one  given 
straight  is  the  join  of  the  pencil- 
points,  then  four  points  and  a 
tangent  at  one  of  them  are  given. 

Thus  by  four  of  its  points  and 
the  tangent  at  one  of  them   a 
*  Pascal,  1640. 


82'.  The  point  on  u  correlated 
to  u^u  is  called  the  '  contact '  on 
u.  Every  other  point  on  u  is  its 
cross  with  a  second  straight  of 
the  conic. 

83'.  On  any  point,  as  F,  on 
any  straight  a  of  the  conic,  its 
second  element  is  its  join  q  with 
the  cross  u(u^, 

84'.  From  the  five  given 
straights  u^  u^^  a,  q,  r^,  oi  K  con- 
struct a  sixth  DD  ^  or  p.  The 
join  d  oi  V  with  the  cross  upt 
and  the  join  di  of  V^  with  the 
cross  u^p  are  copunctal  with  «,. 
Therefore  f  the  three  opposite 
pairs  in  every  complete  set  of 
fans  of  a  hexagram  whose  sides 
are  in  a  conic  concur  in  three 
copunctal  diagonals  whose  bearer 
is  called  a  *  Brianchon  point.' 

This  hexagram  has  sixty  Brian- 
chon points,  since  it  has  sixty 
complete  sets  of  fans. 

85'.  The  cuts  of  the  straights 
of  a  conic  by  any  two  are  pro- 
jective. 

86'.  By  five  of  its  straights  a 
conic  is  completely  determined. 

87'.  Instead  of  five  straights 
may  be  given  the  two  bearers 
and  three  pairs  of  correlated 
points. 

If  one  given  point  is  the  cross 
of  the  bearers,  then  four  straights 
and  a  contact  point  on  one  of 
them  are  given. 

Thus  by  four  of  its  straights 
and  a  contact-point  on  one  of 
f  Brianchon,  1806. 


Art.  8.] 


CURVES   OF    THE    SECOND    DEGREE. 


85 


conic  is  completely  determined. 

88j.  By  three  of  its  points 
and  the  tangents  at  two  of  them 
the  conic  is  completely  deter- 
mined. 

89^.  Interpreting  a  pentastim 
as  a  hexastim  with  two  dots 
coinciding  gives:  In  every  com- 
plete set  of  connectors  of  a  pen- 
tastim whose  dots  are  in  a  conic, 
two  pairs  of  non-consecutive 
connectors  determine  by  their 
two  intersections  a  straight  on 
which  is  the  cross  of  the  fifth 
connector   with   the    tangent   at 


them  a  conic  is  completely  de- 
termined. 

88'.  By  three  of  its  straights 
and  the  contact-points  on  two 
of  them  the  conic  is  completely 
determined. 

89'.  Interpreting  a  pentagram 
as  a  hexagram  with  two  sides 
coinciding  gives:  In  every  com- 
plete set  of  fans  of  a  pentagram 
whose  sides  are  in  a  conic,  two 
pairs  of  non-consecutive  fans 
determine  by  their  two  concurs 
a  point  on  which  is  the  join  of 
the  fifth  fan-point  with  the  con- 
tact-point on  the  opposite  side. 


the  opposite  dot. 

Thence  follows  the  solution  of  the  problems  : 


9O1.  Given  five  points  of  a 
conic,  to  construct  tangents  at 
the  points,  using  the  ruler  only. 

91,.*  The  hexastim  with  a 
pair  of  opposite  connectors  re- 
placed by  tangents  gives:  The 
intersections  of  the  two  opposite 
pairs  in  every  complete  set  of 
connectors  of  a  tetrastim  with 
dots  in  a  conic  are  both  costraight 
with  the  crosses  of  the  two  pairs 
of  tangents  at  opposite  dots. 

Or:  A  tetrastim  with  dots  in 
a  conic  has  each  pair  of  codots 
costraight  with  a  pair  of  fan- 
points  of  the  tetragram  of   tan- 


90'.  Given  five  straights  of  a 
conic,  to  find  contact-points  on 
the  straights,  using  the  ruler  only. 

91'.  The  hexagram  with  a  pair 
of  opposite  fans  replaced  by  con- 
tact-points gives:  The  concurs 
of  the  two  opposite  pairs  in  every 
complete  set  of  fans  of  a  tetra- 
gram with  sides  in  a  conic  are 
both  copunctal  with  the  joins  of 
the  two  pairs  of  contact-points 
on  opposite  sides. 

Or:  A  tetragram  with  sides  in  a 
conic  has  each  pair  of  diagonals 
copunctal  with  a  pair  of  con- 
nectors of  the  tetrastim  of  con- 
gents  at  the  dots.  tacts  on  the  sides. 

The   figure   for    91,   and    that   for  91'   are   identical,   and 
called  Maclaurin's  Configuration.      (See  page  86.) 

92j.  The  tangents  of  a  conic  92'.  The   contact-points  of   a 

range  are  a  conic  pencil.  conic  pencil  are  a  conic  range. 

*  Due  to  Maclaurin,  1748. 


86 


PROJECTIVE    GEOMETRY. 


[Chap.  III. 


93.  The  points  of  a  conic  range  may  now  be  conceived  as 
all  on  a  curve,  a  *  conic  curve,'  their  bearer.     The  straights  of 

the  corresponding  conic  pencil, 
tangents  of  this  conic  range,  may 
now  also  be  conceived  as  all  on 
this  same  conic  curve  on  which 
are  their  contact-points.  Conse- 
quently the  conic  curve  is  dual  to 
itself,  and  so  the  principle  of  dual- 
ity on  a  plane  receives  an  impor- 
tant extension. 

94.  It  follows  immediately  from 
their  generation  that  all  conies  are 
closed  curves,  though  they  may 
be  compendent  through  one  or 
two  points  at  infinity.  With  two 
points  at  infinity  the  curve  is  called 

*  hyperbola  ;*  with  one, '  parabola ;  * 
with  none,  '  ellipse.'  ^ 

95.  If  a  point  has  on  it  tan- 
gents to  the  curve,  it  is  called 
'without*     the     curve;     if    none, 

*  within  '  the  curve.     The  contact- 
point  on  a  tangent  is  '  on  '  the  curve  ;  all  other  points  on  a  tan- 

*  The  generation  shows  that  a  straight  cuts  the  curves  in  two  points  and 
that  from  any  point  two  tangents  to  the  curves  may  be  drawn.  Hence  the 
curves  are  of  the  second  order  and  of  the  second  class,  that  is  they  are  identical 
with  the  conies  of  analytic  geometry.  Analytically  the  equations  -P-j-  XQ  =  o, 
F'  +  A^'  =  o,  where  F,  Q,  P\  Qf  are  linear  functions  of  point  coordinates, 
represent  two  projective  pencils,  the  correlated  rays  corresponding  to  the  same 
value  of  X.  Hence  the  locus  of  the  intersection  of  correlated  rays  is  repre- 
sented by  P(^  —  P' Q  =  o,  a  second-degree  point  equation.  Projective  ranges 
are  represented  by  i?  -f  XS  =  o,  R'  ■\-  XS'  =  o,  where  P,  S,  R\  S'  are  linear 
functions  of  line  coordinates.  The  envelope  of  the  joins  of  correlated  points  is 
represented  by  PS'  —  P' S  =  o,  a  second-degree  line  equation. 

The  projective  generation  of  conies  is  developed  synthetically  in  Steiner's 
Theorie  der  Kegelschnitte,  1866,  and  in  Chasles'  G6ometrie  sup6rieure,  1852. 
For  the  analytic  treatment  see  Clebsch,  Geometrie,  vol.  i,  1876. 


Art.  9.]  pole  and  polar.  87 

gent  are  without  the  curve.  Every  straight  in  its  plane  con- 
tains innumerable  points  without  the  curve,  since  the  straight 
crosses  every  tangent. 

Prob.  13.  Given  four  points  on  a  conic  and  the  tangent  at  one 
of  them,  draw  the  tangent  at  another. 

Prob.  14.  If  the  n  sides  of  a  polygram  rotate  respectively  about 
n  fixed  points  not  costraight,  while  («  — i)  of  a  complete  set  of  fan- 
points  glide  respectively  on  {n  —  \)  fixed  straights,  then  every  remain- 
ing fan-point  describes  a  conic* 

Prob.  15.  In  any  tristim  with  dots  on  a  conic  the  three  crosses 
of  the  connectors  with  the  tangents  at  the  opposite  dots  are 
costraight. f 

Prob.  16.  If  two  given  angles  rotate  about  their  fixed  vertices 
so  that  one  cross  of  their  sides  is  on  a  straight,  either  of  the  other 
three  crosses  describes  a  conic.  J 

Prob.  17.  Construct  a  hyperbola  from  three  given  points,  and 
straights  on  its  figurative  points. 

Art.  9.     Pole  and  Polar. 

96.  Taking  every  tangent  to  a  conic  as  the  dual  to  its  own 
contact-point  fixes  as  dual  to  any  given  point  in  the  plane  one 
particular  straight,  its  '  polar,'  of  which  the  point  is  the 
*pole.' 

97.  With  reference  to  any  given  conic,  to  construct  the 
polar  of  any  given  point  in  its  plane.  Put  on  the  given  point 
Z  two  secants  crossing  the  curve,  one  at  A  and  D,  the  other  at 
B  and  C.  The  join  of  the  other  codots  JT  and  Fof  ABCD  is 
the  polar  of  Z.  Varying  either  secant,  as  ZBC,  does  not 
change  this  polar,  since  on  it  must  always  be  the  cross  5  of 
the  tangents  at  A  and  D,  and  also  the  point  which  D  and  A 
harmonically  separate  from  Z  (given  by  each  of  the  variable 
tetrastims  BXCY). 

98.  The  join  of  any  two  codots  of  a  tetrastim  with  dots  on 
a  conic  is  the  polar  of  the  third  codot  with  respect  to  that 

*  Due  to  Braikenridge,  1735. 
f  From  Pascal ;  dual  from  Brianchon. 

X  Given  by  Newto^  in  Principia,  Book  I,  lemma  xxi,  under  the  name  of 
**  the  organic  description  "  of  a  conic. 


88  PROJECTIVE    GEOMETRY.  [ChAP.  III. 

conic,  and  either  codot  is  the  pole  of  the  join  of  the  other 
two.  Any  point  is  harmonically  separated  from  its  polar  by 
the  conic. 

99.  To  draw  with  ruler  only  the  tangents  to  a  conic  from 
a  point  without,  join  it  to  the  crosses  of  its  polar  with  the 
conic. 

loOj.    Two   points    are   called  100'.  Two  straights  are  called 

*  conjugate '  with  reference  to  a  'conjugate'  with  reference  to  a 

conic  if  one  (and  so  each)  is  on  conic  if  one  (and  so  each)  is  on 

the  polar  of  the  other.  the  pole  of  the  other. 

loij.  All  points  on  a  tangent  loi'.  All   straights   on   a  con- 

are  conjugate  to  its  contact-  tact-point  are  conjugate  to  its 
point.  tangent. 

102,.   The   points   of   a  range  102'.   The   straights  of  a  flat 

are  projective  to  their  conjugates  pencil  are  projective  to  their 
on  its  bearer.  conjugates  on  its  bearer. 

io3j.  With  reference  to  a  given  103'.  With  reference  to  a  given 

conic,      the     '  kerncurve,'      the  conic,  the  *  kerncurve,'  the  poles 

polars  of  all  points  on  a  second  of  all  tangents  on  a  second  conic 

conic  make  a  conic  pencil,  whose  make  a  conic  range,  whose  bearer 

bearer    is    the    *  polarcurve  '    of  is  the  '  polarcurve  *  of  the  second 

the  second  conic.  conic. 

Prob.  18.  Either  diagonal  of  a  circumscribed  tetragram  is  the 
polar  of  the  cross  of  the  others. 

Prob.  19.  A  pair  of  tangents  from  any  point  on  a  polar  harmoni- 
cally separate  it  from  its  pole. 

Prob.  20.  A  pair  of  tangents  are  harmonic  conjugates  with  respect 
to  any  pair  of  straights  on  their  cross  which  are  conjugate  with 
respect  to  the  conic. 

Art.  10.    Involution. 

104.  If  in  a  primal  figure  of  four  elements  (a  *  throw  ')  first 
any  two  be  interchanged,  then  the  other  two,  the  result  is  pro- 
jective to  the  original. 

[That  is,  ABCn  a  BADC  7\  CDAB  a  DCBA:\ 
Let  A  BCD  be  a  throw  on  m.     Project  it  from  V.     Cut  this 
eject  by  a  straight  {m')  on   A.     The  cut  WAB'CD'.      Now 
project  ABCD  from  C     The  cut  of  this  latter  eject  by  V  B  is 


Art.  10.]  INVOLUTION.  89 

B'BVH.     Project  ^'^F/^  from/?  and  cut  the  eject  by  w'.    The 
cut  is  B'AD'd  which  is  perspective  to  BADC. 


105.  Two  projective  primal  figures  of  the  same  kind  of  ele- 
ments and  both  on  the  same  bearer  are  called  *  conjective.' 
When  in  two  conjective  primal  figures  one  particular  element 
has  the  same  mate  to  whichever  figure  it  be  regarded  as  be- 
longing, then  every  element  has  this  property. 

If  AA'BB'  is  projective  to  A'AB'X,  then  by  §  104,  AA'BB' 
is  projective  to  AA' XB' ,  and  having  three  elements  self-corre- 
lated, they  are  identical. 

106.  Two  conjective  figures  such  that  the  elements  are 
mutually  paired  ('  coupled  ')  form  an  *  Involution.*  For  exam- 
ple, the  points  of  a  range,  and,  on  the  same  bearer,  their  con- 
jugates with  respect  to  a  conic,  form  an  involution.  Every 
eject  and  every  cut  of  an  involution  is  an  involution. 

107.  When  two  ranges  are  projective,  the  point  at  infinity 
of  either  one  is  correlated  to  a  point  of  the  other  called  its 
*  vanishing  point.' 

108.  When  two  conjective  ranges  form  an  involution  the 
two  vanishing  points  coincide  in  a  point  called  the  *  center  *  of 
the  involution. 

109.  If  two  figures  forming  an  involution  have  self-corre- 
lated elements,  these  are  called  the  *  double  '  elements  of  the 
involution.  An  involution  has  at  most  two  double  elements  ; 
for  were  three  self-correlated,  all  would  be  self-correlated. 

1 10.  If  a  primal  figure  of  four  elements  is  projective  with 
a  second  made  by  interchanging  two  of  these  elements,  they 
harmonically  separate  the  other  two. 

For  project  the  range  A  BCD  from  Fand  cut  the  eject  by  a 


90 


PROJECTIVE    GEOMETRY. 


[Chap,  III. 


Straight  on  A.     The  cut  AB'CD'   is    projective   to   ABCD, 

which  by  hypothesis  is  projec- 
tive to  ADCB.  Therefore 
ADCB  is  perspective  to 
AB'CD'.  So  vac  is  on  the 
cross  X  of  the  joins  DB'  and 
BD',     So  B  and  D  are  codots 

of  the  tetrastim  VD'XB',  while  A  and  C  are  on  the  connectors 

through  C,  the  third  codot. 

111.  If  an  involution  has  two  double  elements  these  sepa- 
rate harmonically  any  two  coupled  elements.  Let  A  and  C  be 
the  double  elements.  Then  ABCB'  is  projective  to  AB'CB  ; 
therefore  by  §  no  ABCB'  is  harmonic. 

112.  An  involution  is  completely  determined  by  two  couples. 
For  the  projective  correspondence  AA'B  .  .  .  7\  A'AB'  ...  is 
completely  determined  by  the  three  given  pairs  of  correlated 
elements,  and  since  among  them  is  one  couple,  so  are  all  corre- 
lated elements  couples. 

113.  When  there  are  double  elements,  then  the  elements 
of  no  couple  are  separated  by  those  of  another  couple.  In- 
versely, when  the  elements  of  one  couple  separate  those  of 
another,  then  the  elements  of  every  couple  are  separated  by 
those  of  every  other,  and  there  are  no  double  elements. 

114'.  The  three  pairs  of  op- 
posite fan-points  of  a  tetragram 
are  projected  from  any  projec- 
tion-vertex by  three  couples  of 
an  involution  of  straights. 

ii4i.  The  three  pairs  of  op- 
posite connectors  of  a  tetrastim 
are  cut  by  any  transversal  in 
three  couples  of  a  point  involu- 
tion.* 


*Due  to  Desargues,  1639. 


Art.  11.]  PROJECTIVE    CONIC    RANGES.  91 

Let  QRST  be  a  tetrastim  of  which  the  pairs  of  opposite 
connectors  RT  diud  QS,  ST  and  QR,  QT  and  i?5  are  cut  by 
any  transversal  respectively  in  A  and  A',  B  and  B',  (7  and  C. 
From  the  projection-vertex  Q,  the  ranges  ATPR  and  ACA'B' 
are  perspective.  But  A  TPR  and  ABA'C  are  perspective  from 
S.  Therefore  ACA'B'  is  projective  to  ABA'C\  and  therefore 
to  A' CAB  (§  104).  Since  thus^  and  A'  are  coupled,  so  (§  105) 
are  B  and  B\  and  C  and  6". 

115.  To  construct  the  sixth  point  C  oi  an  involution  of 
which  five  points  are  given,  draw  through  C  any  straight,  on 
which  take  any  two  points  Q  and  T.  Join  A  7",  B'Q  crossing 
at  R.  Join  BT,  A' Q  crossing  at  5.  The  join  RS  cuts  the 
bearer  of  the  involution  in  C . 

Prob.  21.  Find  the  center  O  oi  a.  point  involution  of  which  two 
couples  AA'BB'  are  given. 

Prob.  22.  If  two  points  M  and  N  on  m  are  harmonically  sepa- 
rated by  two  pairs  of  opposite  connectors  of  a  tetrastim,  then  so  are 
they  by.  the  third  pair. 

Prob.  23.  To  construct  a  conic  which  shall  be  on  three  given 
points,  and  with  regard  to  which  the  couples  of  points  of  an  involu- 
tion on  a  given  straight  shall  be  conjugate  points. 

Art.  11.    Projective  Conic  Ranges. 

116.  Four  points  on  a  conic  are  called  harmonic  if  they 
are  projected  from  any  (and  so  every)  fifth  point  on  the  conic 
by  four  harmonic  straights. 

117.  A  conic  and  a  primal  figure  or  two  conies  are  called 
projective  when  so  correlated  that  every  four  harmonic  ele- 
ments of  the  one  correspond  to  four  harmonic  elements  of  the 
other. 

118.  If  a  conic  range  and  a  flat  pencil  are  projective,  and 
every  element  of  the  one  is  on  the  correlated  element  of  the 
other,  they  are  called  perspective.  A  conic  is  projected  from 
every  point  on  it  by  a  flat  pencil  perspective  to  it.  Inversely 
the  pencil-point  of  every  flat  pencil  perspective  to  a  conic  is 
on  the  conic. 


"92  PROJECTIVE    GEOMETRY.  [Chap.  III. 

119.  Two  conies  are  projective  if  flat  pencils  respective!}' 
perspective  to  them  are  projective.  Therefore  any  three 
elements  in  one  can  be  correlated  to  any  three  elements  in 
the  other,  but  this  completely  pairs  all  the  elements. 

120.  Two  different  conic  ranges  on  the  same  bearer  have 
at  most  two  self-correlated  elements. 

121.  Two  different  coplanar  conic  ranges  with  a  point  V 
in  common  are  projective  if  every  two  points  costraight  with 
V  are  correlated.     For  both  are  then  perspective  to  the  flat 

pencil  on  V.  Every  common  point  other  than  V  is  self-corre- 
lated ;  but  V  only  when  they  have  there  a  common  tangent. 
They  can  have  at  most  three  self-correlated  points. 

122.  If  a  flat  pencil  V  and  conic  range  k  are  coplanar  and 
projective  but  not  perspective,  then  at  most  three  straights  of 
the  pencil  are  on  their  correlated  points  of  the  conic ;  but  at 
least  one. 

For  any  flat  pencil  M  perspective  to  k  is  projective  to  F, 
and  with  it  determines  in  general  a  second  conic  range  which 
must  have  in  common  with  k  every  point  which  lies  on  its 
correlated  straight  of  V.  So  if  more  than  three  straights  of  V 
were  on  their  correlated  points  of  k,  the  conies  would  be  iden- 
tical and  F  perspective  to  /^. 

Again,  since  every  conic  is  compendent,  and  so  divides  its 
plane  into  two  severed  pieces,  therefore  the  two  different  conies 
if  they  cross  at  their  common  point  M  must  cross  again,  say 
at  P.  In  this  case  the  straights  VP  and  MP  are  correlated, 
and  so  VP  is  on  the  point  P  correlated  to  it  on  k. 

In  case  they  do  not  cross  at  their  common  point  M,  the 
straight  FJ/ corresponds  to  the  common  tangent  at  M,  and  so 
to  the  point  M  correlated  to  it  on  k. 

123.  Two  projective  conic  ranges  on  the  same  curve  form 
an  involution  if  a  pair  of  points  are  doubly  correlated.  Besides 
the  couple  AA^,  let  B  and  B^  be  any  other  two  correlated 
points,  so  that  AA^B  corresponds  to  A^AB^,  The  cross  of 
AA^  and  BB^  call  U,  and  its  polar  u.     Project  AA^B  from  B^, 


Art.  11.]  PROJECTIVE    CONIC    RANGES.  93^ 

Project  A,AB,  from  B.  The  ejects  B,{AA,B)  and  B{A,AB,} 
are  projective,  and  having  the  straight  B^B  (or  BB^)  self-corre- 
lated, so  are  perspective.  The  crosses  of  their  correlated  ele- 
ments are  therefore  costraight.  But  the  cross  of  B^A  with  its 
correlated  straight  BA^  is  known  to  be  on  u,  the  polar  of  U,  the 


cross  of  AA^  with  BB^.  Likewise  the  cross  of  B^A^  with  BA 
is  on  7^.  Therefore  the  point  (7^  correlated  to  C  is  the  cross 
of  CU  with  the  curve.     So  C  and  C^  are  coupled. 

124.  If  two  conic  ranges  form  an  involution,  the  joins  of 
coupled  poiints  are  all  copunctal  on  the  '  invojutioncenter.' 

125.  Calling  projective  the  conic  pencils  dual  to  projective 
conic  ranges,  if  these  ranges  form  an  involution,  so  do  the 
pencils,  and  the  crosses  of  coupled  tangents  are  all  costraight 
on  the  *  involutionaxis.' 

So  two  conic  pencils  forming  an  involution  are  cut  by  each 
of  their  straights  in  two  ranges  forming  an  involution.  Two 
conic  ranges  forming  an  involution  are  projected  from  each  of 
their  points  in  two  flat  pencils,  forming  an  involution. 

126.  If  the  involutioncenter  lies  without  the  conic  bearer 
of  an  involution,  it  has  two  double  elements  where  it  is  cut  by 
the  involutionaxis. 

127.  To  construct  the  self-correlated  points  of  two  pro- 
jective  conic  ranges  on  the  same  conic. — Let  A,  By  C  be  any 
three  points  of  k,  and  A^,  B^,  C^  their  correlated  points  of  k^. 
The  projective  flat  pencils  A{A,B^C,)  and  A^ABC)  have  AA, 
self- corresponding,  hence  they  are  perspective  to  a  range  on 
the  join  u  of  the  cross  of  AB^  and  A^B  with  the  cross  of  AC^ 


94  PROJECTIVE    GEOMETRY.  [Chap.  III. 

and  A^C.     The  crosses  of  the  conic  and  this  join  u  are  the 
self-correlated  points  of  k  and  k^. 

128.  If  the  dots  of  a  tetrastim  are  on  a  conic,  the  six  points 
where  a  straight  not  on  a  dot  cuts  the  conic  and  two  pairs  of 
opposite  connectors  form  an  involution. 

For  the  two  flat  pencils  in  which  the  two  crosses  of  w 
with  the  conic,  P,  P,,  and  two  opposite  dots  R,  T,  are  pro- 
jected from  the  other  two  dots  Q,  S,  are  projective,  and  con- 
sequently so  are  the  cuts  of  these  flat  pencils  by  m;  that  is, 
PBP.A  A  PA,P,B,.  But  PA,P,B,  a  P,B,PA,.  Therefore 
PBP.A  A  P,B,PA,. 

i29j.  Conies  on  which  are  the  129'.   Copunctal   tangents    to 

dots  of  a  tetrastim  are  cut  by  a  conies  on  which  are  the  sides  of 

transversal  in  points  of  an  involu-  a  tetragram  form  an  involution, 

tion.     At  its  double  points  the  The  double  straights  touch  two 

transversal  is  tangent  to  two  of  of   those   conies   at  the   pencil- 

those  conies.  point. 

Prob.  24.  The  pairs  of  points  in  which  a  conic  is  cut  by  the 
straights  of  a  pencil  whose  pencil-point  is  not  on  the  conic  form  an 
involution. 

Art.  12.    Center  and  Diameter. 

130.  The  harmonic  conjugate  of  a  point  at  infinity  with 
respect  to  the  end  points  of  a  finite  sect  is  the  *  center '  of  that 
sect. 

131.  The  pole  of  a  straight  at  infinity  with  respect  to  a 
certain  conic  is  the  *  center '  of  the  conic. 

132.  The  polar  of  any  figurative  point  is  on  the  centre  of 
the  conic,  and  is  called  a  *  diameter.* 

133.  If  a  straight  crosses  a  conic  the  sect  between  the 
crosses  is  called  a  *  chord.* 

The  center  of  a  conic  is  the  center  of  all  chords  on  it. 

134.  The  centers  of  chords  on  straights  conjugate  to  a 
diameter  are  all  on  the  diameter. 

135.  Two  diameters  are  conjugate  when  each  is  the  polar 
of  the  figurative  point  on  the  other. 


Art.  12.]  CENTER    AND    DIAMETER.  95 

136.  The  tangents  at  the  crosses  of  a  straight  with  a  conic 
cross  on  the  diameter  which  is  a  conjugate  to  that  straight. 

137.  The  joins  of  any  point  on  the  conic  to  the  crosses  of  a 
diameter  with  the  conic  are  parallel  to  two  conjugate  diameters. 

138.  Of  two  conjugate  diameters,  each  is  on  the  centers  of 
the  chords  parallel  to  the  other ;  and  if  one  crosses  the  conic, 
the  tangents  at  its  crosses  are  parallel  to  the  other  diameter. 

139.  The  center  of  an  ellipse  is  within  it,  for  its  polar  does 
not  meet  the  curve,  and  so  there  are  no  tangents  from  it  to  the 
curve.  The  centre  of  a  parabola  is  the  contact  point  of  the 
figurative  straight.  The  centre  of  a  hyperbola  lies  without  the 
curve,  since  the  figurative  straight  crosses  the  curve.  The  tan- 
gents from  the  center  to  the  hyperbola  are  called  *  asymptotes.' 
Their  contact-points  are  the  two  points  at  infinity  on  the 
curve. 

140.  If  a  diameter  which  cuts  the  curve  be  given,  the  tan- 
gents at  its  crosses  can  be  constructed  with  ruler  only,  and  so 
however  many  chords  on  straights  conjugate  to  the  diameter. 

141.  Every  fiat  pencil  is  an  involution  of  conjugates  with 
respect  to  a  given  conic.  Hence  the  pairs  of  conjugate  diam- 
eters of  a  conic  form  an  involution. 

If  the  conic  is  a  hyperbola,  the  asymptotes  are  the  double 
straights  of  the  involution.  Hence  any  two  conjugate  diam- 
eters of  a  hyperbola  are  harmonically  separated  by  the  asymp- 
totes ;  and  since  the  hyperbola  lies  wholly  in  one  of  the  two 
explemental  angles  made  by  the  asymptotes,  one  diameter 
cuts  the  curve,  the  other  does  not. 

142.  Any  one  pair  of  conjugate  diameters  of  an  ellipse  is 
always  separated  by  any  other  pair.  Any  one  pair  of  conjugate 
diameters  of  a  hyperbola  is  never  separated  by  any  other  pair. 

143.  If  a  tangent  to  a  hyperbola  cuts  the  asymptotes  at  /i 
and  Cy  then  the  contact-point  B  is  the  center  of  the  sect  AC, 
since  the  tangent  cuts  the  harmonic  pencil  made  by  the  diame- 
ter through  B,  the  conjugate  diameter  and  the  asymptotes,  in 
the  harmonic  range  ABCD  where  D  is  at  infinity.     Just  so  the 


96  PROJECTIVE    GEOMETRY.  [Chap.  IIL 

center  of  any  chord  is  the  center  of  the  costraight  sect  bounded 
by  the  asymptotes. 

144.  If  a  point  is  the  center  of  two  chords  it  is  the  center 
of  the  conic,  for  its  polar  is  the  figurative  straight. 

145.  As  many  points  as  desired  of  a  conic  may  be  con- 
structed by  the  ruler  alone. 

With  the  aid  of  one  fixed  conic  all  problems  solvable  by 
ruler  and  compasses  can  be  solved  by  ruler  alone,  that  is,  by 
pure  projective  geometry.  For  example  :  Of  two  projective 
primal  figures  (say  ranges)  on  the  same  bearer,  given  three 
pairs  of  correlated  elements  to  find  the  self-corresponding  ele- 
ments, if  there  be  any.  Project  the  two  ranges  from  any  point 
V  of  the  given  conic.  These  ejects  are  cut  by  the  conic  in 
projective  conic  ranges.  Of  these  determine  the  self-correlated 
points  by  §  127. 

Project  these  from  V.  The  ejects  cut  the  bearer  of  the 
original  ranges  in  the  required  self-correlated  points. 

Prob.  25.  Find  the  crosses  of  a  straight  with  a  conic  given  only 
by  five  points. 

Prob.  26.  Given  a  conic  and  its  center,  find  a  point  B  such  that 
for  two  given  points  A,  C,  the  center  of  the  sect  AB  shall  be  C. 

Prob.  27.  The  join  of  the  other  extremities  of  two  coinitial  sects 
is  parallel  to  the  join  of  their  centers. 

Prob.  28.  In  an  ellipse  let  A  and  B  be  crosses  of  conjugate  diam- 
eters CAy  CB  with  the  curve.  Through  A'  the  cross  of  the  diameter 
conjugate  to  CA  with  the  curve  draw  a  parallel  to  the  join  AB.  Let 
it  cut  the  curve  again  at  B'.  Then  CB^  is  the  diameter  conjugate 
to  CB. 

Art.  13.    Plane  and  Point  Duality. 

i46j.  On  a  plane  are  00' points,  146'.  On  a  point  are  00 '^  planes, 

a  *  point-field.'  a  'plane-sheaf.' 

i47j.  The  00'  planes  of  a  sin-  147'.  The  00*  points  of  a  sin- 
gle axial  pencil  have  on  them  all  gle  range  have  on  them  all  the 
the  points  of  point-space;  so  planes  of  plane-space;  so  there 
there  are  just  00^  points.  are  just  00'  planes. 

Point-space  is  tridimensional.  Plane-space  is  tridimensional. 


Art.  13] 


PLANE    AND    POINT    DUALITY. 


97 


148.  With  the  straight  as  element,  space  is  of  four  dimen- 
sions. 


On  a  plane  are  00^  straights, 
a  'straight-field.' 

On  a  straight  are  00'  planes, 
and  so  00^  straights. 

On  each  of  the  00^  points  on 


On  a  point  are  00 '^  straights,  a 
*  straight-sheaf.' 

On  a  straight  are  00'  points, 
and  so  00'  straights. 

On  each  of  the  00"  planes  on 


a  plane  are  the  00'  straights  of  a     a  point  are  the  00^  straights  of  a 


straight-sheaf;  so  there  are  just 
CO*  straights. 

149,.  Two  planes  determine  a 
straight,  their  meet. 

150,.  Two  planes  determine  an 
axial-pencil  on  their  meet. 

15 1  J.  Two  bounding  planes 
determine  an  axial  angle. 

152,.  A  plane  and  a  straight 
not  on  it  determine  a  point,  their 
pass. 

i53i.  An  axial  pencil  and  a 
plane  not  on  its  bearer  deter- 
mine a  flat  pencil. 

i54j.  Three  planes  determine 
a  point,  their  apex. 

155,.  Three  planes  determine 
a  plane-sheaf. 

156,.  Two  coplanar  straights 
are  copunctal. 


straight-field;  so  there  are  just 
00*  straights. 

149'.  Two  points  determine  a 
straight,  their  join. 

150'.  Two  points  determine  a 
range  on  their  join. 

151'.  Two  bounding  points 
determine  a  sect. 

152'.  A  point  and  a  straight 
not  on  it  determine  a  plane. 

153'.  A  range  and  a  point  not 
on  its  bearer  determine  a  flat 
pencil. 

154'.  Three  points  determine 
a  plane,  their  junction. 

155'.  Three  points  determine 
a  point-field. 

156'.  Two  copunctal  straights 
are  coplanar. 


157.  Any  figure,  or  the  proof  of  any  theorem  of  configu- 
ration and  determination,  gives  a  dual  figure  or  proves  a  dual 
theorem  by  simply  interchanging  point  with  plane.  Thus  all 
the  pure  projective  geometry  on  a  plane  may  be  read  as  geom- 
etry on  a  point. 

Prob.  29.  If  of  straights  copunctal  in  pairs  not  all  are  copunctal, 
then  all  are  coplanar. 

Prob.  30.  On  a  given  point  put  a  straight  to  cut  two  given  straights. 

Prob.  31.  If  two  triplets  of  planes  oc^y,  a'jS'y'  are  such  that 
the  meets  /3y  and  fSy\  ya  and  y'a\  aft  and  a' ft'  lie  on  three 
planes  a'\  ft'\  y"  which  are  costraight,  then  the  meets  aa\  ftft\ 
yy'  are  coplanar. 


98  PROJECTIVE    GEOMETRY.  [Chap.  III. 

Prob.  32.  Describe  the  figures  in  space  dual  to  the  polystim  and 
the  polygram. 

Art.  14.    Ruled  Quadric  Surfaces. 

158.  The  joins  of  the  correlated  points  of  two  projective 
ranges  whose  bearers  are  not  coplanar  form  a  *  ruled  system  ' 
of  straights  no  two  coplanar.  For  were  two  coplanar,  then 
two  points  on  the  bearer  m  and  two  on  the  bearer  m^  would 
all  four  be  on  this  plane,  and  so  m  and  m^  coplanar,  contrary 
to  hypothesis. 

159.  Let  the  straights  n,  n^,  n^  be  any  three  of  the  elements 
of  a  ruled  system,  and  iV,  any  point  on  n^.  Put  a  plane  on  N^ 
and  the  straight  n.^^  and  let  its  pass  with  n  be  called  N.  The 
straight  iWV,  cuts  n,  n^,  «,  all  three.  Projecting  the  generating 
ranges  of  the  ruled  system  (on  the  bearers  m  and  m^  from  the 
straight  iWV,  (or  m^  as  axis  produces  two  projective  axial 
pencils,  which  having  three  planes  m^n,  m^n^,  m^n^  self-corre- 
sponding, are  identical.  Therefore  every  pair  of  correlated 
points  of  the  ranges  on  m  and  m^  is  coplanar  with  m^ ;  that 
is,  7«5  cuts  every  element  of  the  ruled  system. 

By  varying  the  point  N^  00*  straights  are  obtained,  all  cutting 
all  the  00*  straights  of  the  original  ruled  system  and  making 
on  every  two  projective  ranges.  Of  the  straights  so  obtained 
no  two  cross,  for  that  would  make  two  of  the  first  ruled  system 
coplanar. 

Either  of  these  two  systems  may  be  considered  as  generating 
a  '  ruled  surface,*  which  is  the  bearer  of  both.  Each  of  the 
two  systems  is  completely  determined  by  any  three  straights 
of  the  other,  and  therefore  so  is  the  ruled  surface  also.  From 
the  construction  follows  that  the  straights  of  either  ruled 
system  cut  all  the  straights  of  the  other  in  projective  ranges. 
So  any  two  straights  of  either  system  may  be  considered  as 
bearers  of  projective  ranges  generating  the  other  system,  or 
indeed  the  ruled  surface. 

160.  On  each  point  of  this  ruled  surface  are  two  and  only 
two  straights  lying  wholly  in  the  surface  (one  in  each  ruled 


^j^T    14  ]  RULED    QUADRIC    SURFACES.  99 

system).     So  a  plane  on  one  straight  of  the  ruled  surface  is 

also   on   another  straight  of  this 

surface. 

i6i.  If  in  the  two  generating 
projective  ranges  the  point  at 
infinity  of  one  is  correlated  to  the 
point  at  infinity  of  the  other,  the 
ruled  surface  is  called  a  'hyper- 
bolic-paraboloid.* 

The   join    of   these  figurative 
points  is  on  the  figurative  plane. 
Therefore   the   plane   at  infinity 
cuts  the  surface  in  a  straight  and  so  has  a  second  straight  in 
common  with  the  ruled  surface. 

That  a  hyperbolic-paraboloid  has  two  straights  in  common 
with  the  plane  at  infinity  may  also  be  proved  as  follows : 

Call  the  bearers  of  the  generating  ranges  m  and  w, ,  and  let 
«,  «,  be  any  two  elements,  and /the  element  at  infinity.  By 
§  159  the  ruled  surface  may  be  considered  as  generated  by  the 
straights  on  the  three  elements  «,  «, ,  /.  But  all  these  straights 
must  be  parallel  to  the  same  plane,  namely,  to  any  plane  on  /. 
On  y  and  each  one  of  these  straights  put  a  plane  ;  these  planes 
make  a  parallel-axial-pencil,  and  cut  any  two  of  the  original 
elements  in  projective  ranges  with  the  figurative  points  corre- 
lated. Therefore  the  figurative  straight  joining  the  figurative 
points  of  n  and  w,  is  wholly  on  the  ruled  surface. 

162.  From  §  161  follows  that  all  straights  pertaining  to  the 
same  ruled  system  on  a  hyperbolic-paraboloid  are  parallel  to 
the  same  plane.  Such  planes  are  called  *  asymptote-planes.* 
A  hyperbolic-paraboloid  is  completely  determined  by  two  non- 
coplanar  straights  and  an  asymptote-plane  cutting  them.  To 
get  an  element  cut  the  two  given  straights  by  any  plane  par- 
allel to  the  asymptote-plane,  and  join  the  meets. 

163.  Three  non-crossing  straights,  all  parallel  to  the  same 
plane,  completely  determine  a  hyperbolic-paraboloid.  Let  m, 
Wj,  w,  be  the  given  straights.     The  passes  of  planes  on   m^ 


100 


PROJECTIVE    GEOMETRY. 


[Chap.  IIL 


with  m  and  m^  are  projective  ranges  vi^hose  joins  are  a  ruled 
system. 

But  from  the  hypothesis  one  of  these  planes  is  parallel  to 
both  in  and  m. .  Therefore  their  points  at  infinity  are  corre- 
lated and  the  ruled  surface  is  a  hyperbolic-paraboloid. 

164.  If  two  non-coplanar  projective  ranges  be  each  axially 
projected  from  the  bearer  of  the  other,  two  projective  axial 
pencils  are  formed,  with  those  planes  correlated  on  which  are 
the  correlated  points  of  the  ranges.  If  A,  A^  be  correlated 
points,  then  the  straight  AA^  is  the  meet  of  correlated  planes. 
Thus  two  projective  axial  pencils  with  axes  not  coplanar  gen- 
erate a  ruled  system.  If  the  whole  figure  be  cut  by  a  plane, 
this  will  cut  these  axial  pencils  in  two  projective  flat  pencils,, 
and  the  conic  generated  by  these  will  be  the  cut  of  the  ruled 
surface.  So  every  plane  cuts  it  in  a  conic  or  a  pair  of  straights. 
Hence  no  straight  not  wholly  on  the  surface  can  cut  it  in  more 
than  two  points.  The  surface  is  therefore  of  the  second  degree 
(quadric). 

If  the  plane  at  infinity  cuts  the  ruled  surface  in  a  pair  of 
straights,  it  is  a  hyperbolic-paraboloid.     If  not,  it  is  called   a 

*  hyperboloid  of  one  nappe,*  a  fig- 
ure of  which  is  here  shown. 

l6^\.  Copunctal  straights  par- 
allel to  the  generating  elements  of 
a  hyperboloid  of  one  nappe  are  on 
a  cone.  Copunctal  straights  par- 
allel to  the  generating  elements  of 
a  hyperbolic-paraboloid  are  on  a 
system  of  two  planes. 

For  the  figurative  plane  cuts 
the  hyperboloid  of  one  nappe  in  a 
conic  curve,  but  cuts  the  hyper- 
bolic-paraboloid in  two  straights; 
and  each  of  the  copunctal  straights 
goes  to  a  point  of  the  figurative  cut. 

165.  Each  straight  in  one  ruled  system  of  a  hyperboloid  of 


Art.  14.]  RULED    QUADRIC    SURFACES.  101 

one  nappe  is  parallel  to  one,  but  only  to  one,  straight  in  the 
other  ruled  system.  Of  the  straights  on  a  hyperboHc-parabo- 
loid  no  two  are  parallel.  Let  ti  and  n^ ,  any  two  elements  of 
one  ruled  system,  be  the  bearers  of  the  generating  ranges  R 
and  R^.  If  V  is  the  vanishing  point  of  R,  then  the  straight  on 
F  parallel  to  n^  is  an  element  of  the  other  ruled  system.  But 
for'  the  hyperboHc-paraboloid  V  is  itself  a  figurative  point. 

i66.  Any  straight  of  one  ruled  system  on  a  ruled  surface  is 
called  a  *  guide-straight '  of  the  other  ruled  system. 

1671.  A  ruled  system  is  cut  by  167'.  A  ruled  system  is   pro- 

any  two  of  its  guide-straights  in  jected  from  any  two  of  its  guide- 
projective  ranges.  straights  in  projective  axial  pen- 

cils. 

For  if  m,  ;//,,  m^  be  any  three  guide-straights  of  the  ruled 
system,  the  planes  on  7n^  cut  in  and  rn^  in  projective  ranges  the 
joins  of  whose  correlated  points  are  the  elements  of  the  ruled 
system.  Again,  if  the  points  on  m^  be  projected  axially  from 
m  and  w,,  the  meets  of  the  planes  so  correlated  are  the  ele- 
ments of  the  ruled  system. 

168.  Four  straights  of  a  ruled  system  are  called  harmonic 
straights  if  they  are  cut  in  four  harmonic  points  by  one  (and  so 
by  every)  guide-straight.  By  three  straights,  no  two  coplanar, 
a  fourth  harmonic  is  determined  lying  in  a  ruled  system  with 
the  given  three  and  on  a  fourth  harmonic  point  to  any  three 
costraight  points  of  the  given  three. 

169.  A  plane  cutting  the  ruled  surface  in  a  straight  m  of  one 
ruled  system  and  consequently  also  in  a  straight  n  of  the  other 
ruled  system  has  in  common  with  the  surface  no  point  not  on 
one  of  these  straights.  For  any  straight  from  such  a  point 
cutting  both  these  straights  would  lie  wholly  on  the  ruled  sur- 
face ;  and  so  therefore  would  their  whole  plane,  which  is  im- 
possible. Any  third  straight  coplanar  with  m  and  n  on  their 
cross  has  no  second  point  in  common  with  the  surface  and  so 
is  a  tangent,  and  the  plane  of  m  and  n  is  called  tangent  at  their 
cross,  the  point  nin. 


102  PROJECTIVE    GEOMETRY.  [Chap.  IIL 

The  number  of  planes  tangent  to  the  ruled  surface  and  on 
a  given  straight  equals  the  number  of  points  the  straight  has 
in  Common  with  the  ruled  surface,  that  is  two  ;  so  the  ruled 
surface  is  of  the  second  class. 

170.  Project  the  two  generating  ranges  of  a  ruled  system 
from  any  projection-vertex  Fnot  on  it.  The  eject  consists  of 
two  copunctal  projective  fiat  pencils.  The  plane  of  any  two 
correlated  straights  is  on  an  element  of  the  ruled  system.  All 
such  planes  form  a  cone  of  planes. 

The  points  of  contact  of  these  planes  with  the  ruled  surface 
are  a  conic  range.  The  planes  tangent  to  a  ruled  surface  at 
the  points  on  its  cut  with  a  plane  form  a  cone  of  planes. 

171.  The  cut  of  a  hyperbolic-paraboloid  by  a  plane  not  on 
an  element  has  on  it  the  meets  of  the  plane  with  the  two  figu- 
rative elements,  and  so  is  a  hyperbola  except  when  their  cross 
is  on  the  plane,  in  which  case  it  is  a  parabola.  The  figurative 
plane  is  a  tangent  plane. 

172.  The  planes  tangent  at  the  figurative  points  of  a  hyper- 
boloid  of  one  nappe  are  all  proper  planes,  copunctal  and  form- 
ing  a  cone  of  planes  tangent  to  the  '  asymptote-cone  '  of  the 
hyperboloid.  Each  element  to  the  asymptote-cone  is  parallel 
to  one  element  of  each  ruled  system. 

Any  plane  not  on  an  element  of  the  hyperboloid  of  one 
nappe  cuts  it  in  a  hyperbola,  parabola,  or  ellipse,  according  as 
it  is  parallel  to  two  elements,  one,  or  no  element  of  the  asymp- 
tote-cone, that  is,  according  as  it  has  in  common  with  the  figu- 
rative conic  on  the  hyperboloid  two  points,  one,  or  no  point. 

173.  If  an  axial  pencil  and  a  ruled  system  are  projective, 
they  generate  in  general  a  *  twisted  cubic  curve,'  which  any 
plane  cuts  in  one  point  at  least  and  three  at  most.  For  a 
plane  cuts  the  ruled  system  in  a  conic  range  perspective  to  it, 
of  which  in  general  three  points  at  most  He  on  the  correspond- 
ing planes  of  the  pencil. 

174.  The  ruled  quadric  surface  is  the  only  surface  doubly 


Art.  14.]  ruled  quadric  surfaces.  103 

ruled.     The  figure  of  two  so  united  ruled  systems  is  one  of  the 
most  noteworthy  discovered  by  the  modern  geometry.* 

175.  To  find  the  straights  crossing  four  given  straights. — 
Let  u^,  «,,  «,,  u^  be  the  given  straights.  Projecting  the  range 
R^  on  «,  from  the  axes  «,  and  u^  gives  two  axial  pencils,  each 
perspective  to  /?,,  and  consequently  projective.  The  meets  of 
their  correlated  planes  are  all  the  00*  straights  on  ?/,,  u^,  u^y 
and  form  a  ruled  system  of  which  u^,  u^,  u^  are  guide-straights. 
The  two  projective  axial-pencils  cut  the  fourth  straight  u^  in 
two  *  conjective  '  ranges.  [Two  projective  primal  figures  of  the 
same  kind  and  on  the  same  bearer  are  called  conjective.]  If 
now  a  straight  m  of  the  ruled  system  crosses  u^,  then  the  two 
correlated  planes  of  which  this  straight  m  is  the  meet  must  cut 
u^  in  the  same  point,  which  consequently  is  a  self-correspond- 
ing point  of  the  two  conjective  ranges.  Since  there  are  two 
such  (the  points  common  to  «^  and  the  ruled  surface),  so  there 
are  two  straights  (real  or  conjugate  imaginary)  crossing  four 
given  straights.  Their  construction  is  shown  to  depend  on 
that  for  the  two  self-correlated  points  of  two  conjective  ranges. 

This  important  problem  in  the  four-dimensional  space  of 
straights,  *  what  is  common  to  four  straights  ? '  is  the  analogue 
of  the  problem  in  the  space  of  points,  '  what  is  common  to 
three  points?*  and  its  dual  in  the  space  of  planes,  *  what  is 
common  to  three  planes  ?  * 

It  shows  not  only  their  fundamental  diversity,  .but  also,  as 
compared  to  points-geometry  and  planes-geometry,  the  inher- 
ently quadratic  character  of  straights-geometry. 

Prob.  33.  Find  the  straights  cutting  two  given  straights  and 
parallel  to  a  third. 

Prob.  34.  Three  diagonals  of  a  skew  hexagram  whose  six  sides 
are  on  a  ruled  surface  are  copunctal. 

Prob.  35.  If  a  fiat  pencil  and  a  range  not  on  parallel  planes  are 
projective,  then  straights  on  the  points  of  the  range  parallel  to  the 
correlated  straights  of  the  pencil  form  one  ruled  system  of  a  hyper- 
bolic-paraboloid. 

*See  Monge,  Journal  de  T^cole  poly  technique,  Vol.  I. 


104  PROJECTIVE    GEOMETRY.  [ChaP.  III. 

Prob.  36.  What  is  the  locus  of  a  point  harmonically  separated 
from  a  given  point  by  a  ruled  surface  ? 

Art.  15.    Cross-Ratio. 

176.  Lindemann  has  shown  how  every  one  number,  whether 
integer,  fraction,  or  irrational,  +  or  — ,  may  be  correlated  to 
one  point  of  a  straight,  without  making  any  use  of  measure- 
ment, without  any  comparison  of  sects  by  application  of  a  unit 
sect.*  He  gets  an  analytic  definition  of  the  '  cross-ratio  *  of 
four  copunctal  straights.  Then  this  expression  is  applied  to 
four  costraight  points.  Then  is  deduced  that  the  number  pre- 
viously attached  to  a  point  on  a  straight  is  the  same  as  the 
cross-ratio  of  that  point  with  three  fixed  points  of  the  straight. 
Thus  analytic  geometry  and  metric  geometry  may  be  founded 
without  using  ratio  in  its  old  sense,  involving  measurement. 
Thus  also  the  non-Euclidean  geometries,  that  of  Bolyai-Loba- 
ch^vski  in  which  the  straight  has  two  points  at  infinity,  and 
that  of  Riemann  in  which  the  straight  has  no  point  at  infinity, 
may  be  treated  together  with  the  limiting  case  of  each  between 
them,  the  Euclidean  geometry,  wherein  the  straight  has  one 
but  only  one  point  at  infinity. 

Relinquishing  for  brevity  this  pure  projective  standpoint 
and  reverting  to  the  old  metric  usages  where  an  angle  is  an  in- 
clination, a  sect  is  a  piece  of  a  straight,  and  any  ratio  is  a 
number;  distinguishing  the  sect  AC  from  CA  as  of  opposite 
*  sense,'  so  that  AC=  -  CA,  the  ratio  [AC/BC]/[An/BD]  is 
called  the  cross-ratio  of  the  range  ABCD  and  is  written  {ABCUl 
where  A  and  B,  called  conjugate  points  of  the  cross-ratio,  may 
be  looked  upon  as  the  extremities  of  a  sect  divided  internally 
or  externally  by  C  and  again  by  D.\ 

*  Von  Staudt  in  BeitrSge  zur  Geometric  der  Lage,  1856-60,  determines  the 
projective  definition  of  number,  and  thus  makes  the  metric  geometry  a  conse- 
quence of  projective  geometry. 

f  The  fundamental  property  of  cross-ratio  is  stated  in  the  Mathematical  Col- 
lections of  Pappus,  about  400  B.C.  The  cross-ratio  is  the  basis  of  Poncelet's 
Trait6  des  propri6t6s  projectives,  1822,  which  distinguishes  sharply  the  projec- 
tive and  metric  properties  of  curves. 


JVrT.  15.]  CROSS-RATIO.  '  105 

177.  If  on  ABCD  respectively  be  the  straights  abed  co- 
punctal  on  V,  then  A  C/BC=  A  A  VC/AB  VC 

or  A  C/BC  =  iA  V.  VC  sin  {ac)/^B  V.  VC  sin  (be). 
AD/BD  =  A  A  VD/ABVD 

=  iA  V.  VD  sin  {aa)/iBV.  VD  sin  {bd). 

Therefore       IABCD']  =  [sin  (^^)/sin  (^^)]/[sin  ad/sin  (bd)] . 

Thus  as  the  cross-ratio  of  any  flat  pencil  V[abcd~\  or  axial 
pencil  u{a/3yS)  may  be  taken  the  cross-ratio  of  the  cut  ABCD 
on  any  transversal. 

178.  Two  projective  primal  figures  are  'equicross;'  and 
inversely  two  equicross  primal  figures  are  projective. 

179.  As  D  approaches  the  point  at  infinity,  AD/BD  ap- 
proaches I.  The  cross-ratio  [ABCD]  when  D  is  figurative 
equals  A  C/BC 

180.  Given  three  costraight  points  ABC,  to  find  D  so  that 
[ABCD]  may  equal  a  given  number  n  (+  or  — ).  On  any 
straight  on  C  take  A'  and  B'  such  that  CA' /CB'  =.n\  A'  and 
B'  lying  on  the  same  side  oi  C  \i  n  be  positive,  but  on  opposite 
sides  if  n  be  negative.  Join  AA' ,  BB\  crossing  in  V.  The 
parallel  to  A'B^  on  Fwill  cut  AB  in  the  required  D.  For  if 
D'  be  the  point  at  infinity  on  A'B',  and  ABCD  be  projected 
from  V,  then  A'B'CD'  is  a  cut  of  the  eject ;  so 

[ABCD]  =  [A'B'CD']  =A'C/B'C  =  n. 

181.  If  [ABCD]  =  [ABCD,],  then  D,  coincides  with  D. 

182.  If  two  figures  be  complete  plane  perspectives,  four 
costraight  points  (or  copunctal  straights)  in  one  are  equicross 
with  the  correlated  four  in  the  other.  Let  O  be  the  center  of 
perspective.  Let  M  and  M'  be  any  pair  of  correlated  points 
of  the  two  figures,  A^and  N'  another  pair  of  correlated  points 
lying  on  the  straight  OMM'  whose  cross  with  the  axis  of  per- 
spective is  X.     Then  [OXMN]  =  [6XM'N'\ 

That  is,    [OM/XM]/[ON/XN]  =  [OM' /XM']/[ON' /XN']. 
Therefore  [OM/XM]/[OM' /XM']  =  [ON/XN]/[ON' /XN']. 
That  is,  [OXMM']  =  [OXNN'] ;  or  the  cross-ratio  [OXMM'] 


106  PROJECTIVE    GEOMETRY.  [ChaP.  III. 

is  constant  for  all  pairs  of  correlated  points  M  and  M'  taken 
on  a  straight  OX  on  the  center  of  perspective. 

Next  let  L  and  L'  be  another  pair  of  correlated  points  and 
Y  the  cross  of  OLL  with  the  axis  of  perspective.  Since  LM 
and  L'M'  cross  on  some  point  Z  of  the  axis  XY,  therefore  if 
OXMM'  be  projected  from  Z,  the  cut  of  the  eject  hy  OY  is 
O  YLL',  So  \_6XMM'^,  =  [(9  YLL'^^ ;  or  the  cross-ratio  lOXMM'^ 
is  constant  for  all  pairs  of  correlated  points. 

It  is  called  the  *  parameter '  of  the  correlation.  When  the 
parameter  equals  —  I,  the  range  OXMM'  is  harmonic,  and  two 
correlated  elements  correspond  doubly,  are  coupled,  and  the 
correlation  is  'involutorial.* 

183.  When  the  correlation  is  involutorial  and  the  center  of 
perspective  is  the  figurative  point  on  a  perpendicular  to  the 
axis  of  perspective,  this  is  called  the  *axis  of  symmetry,'  and 
the  complete  plane  perspectives  are  said  to  be  *  symmetrical.' 

184.  When  the  correlation  is  involutorial  and  the  axis  of 
perspective  is  figurative,  then  the  center  of  perspective  is  called 
the  '  symcenter,'  and  the  complete  plane  perspectives  are  said 
to  be  *  symcentral.* 

Prob.  37.  In  a  plane  are  given  a  parallelogram  and  any  sect. 
With  the  ruler  alone  find  the  center  of  the  sect  and  draw  a  parallel 
to  it. 

Prob.  38.  The  locus  of  a  point  such  that  its  joins  to  four  given 
points  have  a  given  cross-ratio  is  a  conic  on  which  are  the  points. 

Prob.  39.  If  the  sides  of  a  trigram  are  tangent  to  a  conic,  the 
joins  of  two  of  its  fan-points  to  any  point  on  the  polar  of  the  third 
are  conjugate  with  respect  to  the  conic. 

Prob.  40.  If  from  any  point  of  the  sect  between  the  contact- 
points  of  a  pair  of  tangents  to  a  parabola  straights  be  drawn  parallel 
to  these  tangents,  the  join  of  their  proper  crosses  with  the  tangents 
will  be  a  tangent. 


Art.  1.]  CORRESPONDENCE    OF    POINTS    ON    CONICS.  101 


Chapter  IV. 
HYPERBOLIC  FUNCTIONS. 

By  James  McMahon, 
Assistant  Professor  of  Mathematics  in  Cornell  University. 

Art.  1.    Correspondence  of  Points  on  Conics. 

To  prepare  the  way  for  a  general  treatment  of  the  hyper- 
boHc  functions  a  preliminary  discussion  is  given  on  the  relations 
between  hyperbolic  sectors.  The  method  adopted  is  such  as 
to  apply  at  the  same  time  to  sectors  of  the  ellipse,  including 
the  circle;  and  the  analogy  of  the  hyperbolic  and  circular 
functions  will  be  obvious  at  every  step,  since  the  same  set  of 
equations  can  be  read  in  connection  with  either  the  hyperbola 
or  the  ellipse."^  It  is  convenient  to  begin  with  the  theory  of 
correspondence  of  points  on  two  central  conics  of  like  species, 
i.e.  either  both  ellipses  or  both  hyperbolas. 

To  obtain  a  definition  of  corresponding  points,  let  (9,^,, 
O^B^  be  conjugate  radii  of  a  central  conic,  and  O^A^ ,  O^B^ 
conjugate  radii  of  any  other  central  conic  of  the  same  species ; 
let  P, ,  P^  be  two  points  on  the  curves;  and  let  their  coordi- 
nates referred  to  the  respective  pairs  of  conjugate  directions 
be  (;r, ,  7,),  (x^ ,  y^\  then,  by  analytic  geometry, 

*  The  hyperbolic  functions  are  not  so  named  on  account  of  any  analogy 
with  what  are  termed  Elliptic  Functions.  '  *  The  elliptic  integrals,  and  thence 
the  elliptic  functions,  derive  their  name  from  the  early  attempts  of  mathemati- 
cians at  the  rectification  of  the  ellipse.  ...  To  a  certain  extent  this  is  a 
disadvantage;  .  .  .  because  we  employ  the  name  hyperbolic  function  to  de- 
note cosh  M,  sinh  u,  etc.,  by  analogy  with  which  the  elliptic  functions  would  be 
merely  the  circular  functions  cos  0,  sin  </>,  etc.  ..."  (Greenhill,  Elliptic 
Functions,  p.  175.) 


108 


HYPERBOLIC    FUNCTIONS. 


[Chap.  IV. 


l^ow  if  the  points  P^ ,  P,  be  so  situated  that 


b,       bl 


(2) 


the  equalities  referring  to  sign  as  well  as  magnitude,  then  P^ , 
P^  are  called  corresponding  points  in  the  two  systems.  If  d  > 
Q^  be  another  pair  of  correspondents,  then  the  sector  and  tri- 


angle P^O^Q^  are  said  to  correspond  respectively  with  the 
sector  and  triangle  P^O^Q^.  These  definitions  will  apply  also 
when  the  conies  coincide,  the  points  P,  ,  P,  being  then  referred 
to  any  two  pairs  of  conjugate  diameters  of  the  same  conic. 

In  discussing  the  relations  between  corresponding  areas  it 
is  convenient  to  adopt  the  following  use  of  the  word  "  measure": 
The  measure  of  any  area  connected  with  a  given  central  conic 
is  the  ratio  which  it  bears  to  the  constant  area  of  the  triangle 
formed  by  two  conjugate  diameters  of  the  same  conic. 

For  example,  the  measure  of  the  sector  A^O^P^  is  the  ratio 
sector  ^,(9,P, 
triangle  A^O.B^ 


Art.  3.]  areas  of  corresponding  sectors.  109' 

and  is  to  be  regarded  as  positive  or  negative  according  as 
A^O^P^  and  A^O^B^  are  at  the  same  or  opposite  sides  of  their 
common  initial  Hne. 

Art.  2.  Areas  of  Corresponding  Triangles. 
The  areas  of  corresponding  triangles  have  equal  measures. 
For,  let  the  coordinates  of  /*,,  Q^  be  (;i^,  ,7,),  (-^Z,  J^/),  and  let 
those  of  their  correspondents/*,,  Q^  be  (^3,  j^,),  (^/, ^Z);  let  the 
triangles /*, (9,(2, ,  Pj 6^,0,  be  7^,,  7",,  and  let  the  measuring  tri- 
angles A^O^B^,  A^O^B^  be  A",,  JT, ,  and  their  angles  (w, ,  &?, ; 
then,  by  analytic  geometry,  taking  account  of  both  magnitude 
and  direction  of  angles,  areas,  and  lines, 

T\  _  iC-^uT/- ^i>.)  sin  6?,  _x,  yj___^yy. 
K^  \aj)^  sin  od^  a^    b^        a^  b^  * 

7\  ^  i{^,y.'-<:y.)  sin  00,  _  ^  ^'  _  f/T?^ 
K^  ia^b^  sin  w,  a,    b^         a,  b^ 

Therefore  ^  =  J/  ^^^ 

Art.  3.  Areas  of  Corresponding  Sectors. 
The  areas  of  corresponding  sectors  have  equal  measures. 
For  conceive  the  sectors  S,,  5,  divided  up  into  infinitesimal 
corresponding  sectors ;  then  the  respective  infinitesimal  corre- 
sponding triangles  have  equal  measures  (Art.  2) ;  but  the 
given  sectors  are  the  limits  of  the  sums  of  these  infinitesimal 
triangles,  hence 

In  particular,  the  sectors  A^O^P^,  AfiJP^  have  equal  meas- 
ures; for  the  initial  points  ^,,  A^  are  corresponding  points. 

It  may  be  proved  conversely  by  an  obvious  reductio  ad 
absurdum  that  if  the  initial  points  of  two  equal-measured 
sectors  correspond,  then  their  terminal  points  correspond. 

Thus  if  any  radii  (9,-^,,  O^A^  be  the  initial  lines  of  two 
equal-measured  sectors  whose   terminal  radii   are  (9,P,,  O^P^y 


110  HYPERBOLIC    FUNCTIONS.  [ChAP.  IV. 

tken  Pj,  P^  are  corresponding  points  referred  respectively  to 
the  pairs  of  conjugate  directions  O^A^,  O^B^,  and  O^A^,  0^B^\ 
that  is, 

^  =  -'     ^  =  -^ 

Prob.  I.  Prove  that  the  sector  F^O^Q^  is  bisected  by  the  line 
joining  O^  to  the  mid-point  of  P ^Ql.  (Refer  the  points  F^^  Q^,  re- 
spectively, to  the  median  as  common  axis  of  x^  and  to  the  two 
opposite  conjugate  directions  as  axis  of  y,  and  show  that  F^,  Q^ 
are  then  corresponding  points.) 

Prob.  2.  Prove  that  the  measure  of  a  circular  sector  is  equal  to 
the  radian  measure  of  its  angle. 

Prob.  3-  Find  the  measure  of  an  elliptic  quadrant,  and  of  the 
sector  included  by  conjugate  radii. 

Art.  4-.    Characteristic  Ratios  of  Sectorial 
Measures. 

Let  A.O^P^  =  5,  be  any  sector  of  a  central  conic;  draw 
P^M^  ordinate  to  (9,^,,  i.e.  parallel  to  the  tangent  at  A^\ 
let  O^M^  =  x^y  M^P^  =^  y^,O^A^—a^,  and  the  conjugate  radius 
O^B^  =  b^ ;  then  the  ratios  xja^ ,  yjb^  are  called  the  charac- 
teristic ratios  of  the  given  sectorial  measure  SJK^.  These 
ratios  are  constant  both  in  magnitude  and  sign  for  all  sectors 
of  the  same  measure  and  species  wherever  these  may  be  situ- 
ated (Art.  3).  Hence  there  exists  a  functional  relation  be- 
tween the  sectorial  measure  and  each  of  its  characteristic 
ratios. 

Art.  5.    Ratios  Expressed  as  Triangle-measures. 

The  triangle  of  a  sector  and  its  complementary  triangle  are 
measured  by  the  two  characteristic  ratios.  For,  let  the  triangle 
A,0,P,  and  its  complementary  triangle  Pfi^B^  be  denoted  by 
r„  r/;  then 

K^      i^i^i  sin  ft?,      b^    \ 

T^'  _  ^b^x^  sin  gj,  _  x^ 
K^      \afi^  sin  ft?,  "~~  a^ 


(s) 


ART.  7.]  FUNCTIONAL    RELATIONS    FOR    ELLIPSE.  Ill 

Art.  6.    Functional  Relations  for  Ellipse. 

The  functional  relations  that  exist  between  the  sectorial 
measure  and  each  of  its  characteristic  ratios  are  the  same 
for  all  eUiptic,  in- 
cluding circular,  sec- 
tors (Art.  4).  LetP,, 
P^  be  corresponding 
points  on  an  ellipse 
and  a  circle,  referred  d 
to  the  conjugate  di-  ^^  -^^2  ^"^ 

rections  (9,^,,  O^B^,  and  O^  A^,O^B^,  the  latter  pair  being  at 
right  angles ;  let  the  angle  A^O^P^  =  0  in  radian  measure;  then 


ia: 


e. 


(6) 


-  =  cos_,     -  =  s.n-^; 


[«,  =  *. 


hence,  in  the  ellipse,  by  Art.  3, 

_>  =  cos  -±,     -^  =  sm  — !-. 


(7) 


Prob.  4.  Given  x^  =  \a^ ;  find  the  measure  of  the  elliptic  sector 
A^OxPx.     Also  find  its  area  when  ^^  =  4,  ^j  =  3,  g?  =  60°. 

Prob.  5.  Find  the  characteristic  ratios  of  an  elliptic  sector  whose 
measure  is  \7t. 

Prob.  6.  Write  down  the  relation  between  an  elliptic  sector  and 
its  triangle.     (See  Art.  5.) 


Art.  7.    Functional  Relations  for  Hyperbola. 

The  functional  relations  between  a  sectorial  measure  and 
its  characteristic  ratios  in  the  case  of  the  hyperbola  may  be 
written  in  the  form 


^  =  cosh  ^' 


•^  =  sinh  -£l. 


k:  b,        k: 

and  these  express  that  the  ratio  of  the  two  lines  on  the  left  is 
a  certain  definite  function  of  the  ratio  of  the  two  areas  on  the 
right.     These  functions  are  called  by  analogy  the  hyperbolic 


112  HYPERBOLIC    FUNCTIONS.  [ChAP.  IV- 

cosine  ana  the  nyperbolic  sine.  Thus,  writing  u  for  SJK^,  the 
two  equations 

—  =  cosh  Uj    "^  =  sinh  u  (SV 

a,  d, 

serve  to  define  the  hyperbolic  cosine  and  sine  of  a  given  secto- 
rial measure  u ;  and  the  hyperbolic  tangent,  cotangent,  secant, 
and  cosecant  are  then  defined  as  follows : 


sinh  u  .  cosh« 

tanh  u  = 7^,     cotn  u  =  -^—r- — , 

cosh  2^  sinh  u 


sech  u  = '-. — ,     csch 

cosh  u 


sinh  u   J 


(9> 


The  names  of  these  functions  may  be  read  *'  h-cosine," 
"  h-sine,"  ''  h-tangent,"  etc. 

Art.  8.  Relations  between  Hyperbolic  Functions. 
Among  the  six  functions  there  are  five  independent  rela- 
tions, so  that  when  the  numerical  value  of  one  of  the  functions 
is  given,  .the  values  of  the  other  five  can  be  found.  Four  of 
these  relations  consist  of  the  four  defining  equations  (9).  The 
fifth  is  derived  from  the  equation  of  the  hyperbola 

<    i>:  ~ 

giving 

cosh'«  —  sinh'«  =  i.  (10) 

By  a  combination  of  some  of  these  equations  other  subsidi- 
ary relations  may  be  obtained;  thus,  dividing  (10)  successively 
by  cosh'  u,  sinh'  u,  and  applying  (9),  give 

I  —  tanh'  u  =  sech'  //,  ) 
coth'  u  —  I  =  csch'  u.  ) 

Equations  (9),  (10),  (11)  will  readily  serve  to  express  the 
value  of  any  function  in  terms  of  any  other.  For  example^ 
when  tanh  u  is  given, 

coth  u  =  — - — ,  sech  u  =  ^  \  —  tanh'  u^ 

tanh  u 


Art.  8.]      relations  between  hyperbolic  functions.  IIS 

tanh  u 


cosh  u  =  —  ,         sinh  u 


^/  I  —  tanh'  u  y   I  —  tanh 


'u 


,  a/  I  —  tanh  u 

csch  u  =  -^ 

tanh  u 

The  ambiguity  in  the  sign  of  the  square  root  may  usually 
be  removed  by  the  following  considerations :  The  functions 
cosh  u,  sech  u  are  always  positive,  because  the  primary  char- 
acteristic ratio  x^a^  is  positive,  since  the  initial  line  O^A^  and 
thp  abscissa  O^M^  are  similarly  directed  from  (9, ,  on  which- 
ever branch  of  the  hyperbola  P^  may  be  situated;  but  the  func- 
tions sinh  u,  tanh  ti,  coth  u,  csch  u,  involve  the  other  charac- 
teristic ratio  jj^i/^i ,  which  is  positive  or  negative  according  as 
jj  and  d^  have  the  same  or  opposite  signs,  i.e.,  as  the  measure 
u  is  positive  or  negative ;  hence  these  four  functions  are  either 
all  positive  or  all  negative.  Thus  when  any  one  of  the  func- 
tions sinh  u,  tanh  u,  csch  «,  coth  u,  is  given  in  magnitude  and 
sign,  there  is  no  ambiguity  in  the  value  of  any  of  the  six 
hyperboHc  functions ;  but  when  either  cosh  u  or  sech  u  is 
given,  there  is  ambiguity  as  to  whether  the  other  four  functions 
shall  be  all  positive  or  all  negative. 

The  hyperbolic  tangent  may  be  expressed  as  the  ratio  of 
two  lines.     For  draw  the  tangent 
Hne^^'^  t\  then 


'=i 

X      a 
''~a~'b' 

I 

X 

a 

t        i 
'a~b' 

The  hyperboHc  tangent  is  the  measure  of  the  triangle  OAC, 
For 

OAC      at       t       ^     .  .    . 

OAB  =  ab=  !  =  '''''''"■  ('3> 

Thus  the  sector  AOP,  and  the  triangles  AOP,  POB,  AOC, 
are  proportional  to  Uy  sinh  u,  cosh  «,  tanh  u  (eqs.  5,  13) :  hence 

sinhw  >  «>  tanhw.  (i4> 


114 


HYPERBOLIC    FUNCTIONS. 


[Chap.  IV. 


Prob.  7.  Express  all  the  hyperbolic  functions  in  terms  of  sinh  u. 
Given  cosh  z^  =  2,  find  the  values  of  the  other  functions: 

Prob.  8.  Prove  from  eqs.  10,  n,  that  coshz/>  sinh  z^,  cosh2^>i, 
tanh  u  <\^  sech  u  <  i. 

Prob.  9.  In  the  figure  of  Art.  i,  let  0A  =  2,  OB=i,  AOB  =  60°, 
and  area  of  sector  AOF  —  3;  find  the  sectorial  measure,  and  the 
two  characteristic  ratios,  in  the  elliptic  sector,  and  also  in  the  hyper- 
bolic sector;  and  find  the  area  of  the  triangle  AOF.  (Use  tables  of 
cos,  sin,  cosh,  sinh.) 

Prob.  10.  Show  that  coth  u,  sech  u,  csch  u  may  each  be  ex- 
pressed as  the  ratio  of  two  lines,  as  follows:  Let  the  tangent  at  F 
make  on  the  conjugate  axes  OA,  OB^  intercepts  OS  =  m,  OT  =  n\ 
let  the  tangent  at  B,  to  the  conjugate  hyperbola,  meet  OF  in  R^ 
making  BR  =  /;  then 

coth  u  —  l/a^     sech  u  =  m/a,     csch  u  =  n/b. 

Prob.  II.  The  measure  of  segment  AMF  is  sinh  u  cosh  u  —  u. 
Modify  this  for  the  ellipse.  Modify  also  eqs.  10-14,  3.nd  probs. 
8,  10. 

Art.  9.    Variations  of  the  Hyperbolic  Functions. 

Since  the  values  of  the  hyperbolic  functions  depend  only 
on  the  sectorial  measure,  it  is  convenient,  in  tracing  their  vari- 
ations, to  consider  only  sectors  of  one 
half  of  a  rectangular  hyperbola,  w^hose 
conjugate  radii  are  equal,  and  to  take  the 
principal  axis  OA  as  the  common  initial 
line  of  all  the  sectors.  The  sectorial 
measure  u  assumes  every  value  from  —  00, 
through  o,  to  -f-  00 ,  as  the  terminal  point 
P  comes  in  from  infinity  on  the  lower 
branch,  and  passes  to  infinity  on  the  upper 
branch  ;  that  is,  as  the  terminal  line  OP 
sv^^ings  from  the  lower  asymptotic  posi- 
tion y  z=  —  X,  to  the  upper  one,  y  =  x.  It  is  here  assumed, 
but  is  proved  in  Art.  17,  that  the  sector  AOP  becomes  infinite 
as  P  passes  to  infinity. 

Since  the  functions  cosh  u,  sinh  u,  tanh  u,  for  any  position 


Art.  9.]  VARIATIONS  OF  THE  HYPERBOLIC  FUNCTIONS.  115 

of  OP^  are  equal  to  the  ratios  of  x^  y,  t,  to  the  principal  radius 
^,  it  is  evident  from  the  figure  that 

cosh  0=1,     sinh  0  =  0,     tanh  0=0,  (15) 

and  that  as  u  increases  towards  positive  infinity,  cosh  u,  sinh  u 
are  positive  and  become  infinite,  but  tanh  u  approaches  unity 
as  a  limit ;  thus 

cosh  00  =  00  ,     sinh  00  =  00  ,     tanh  00  =  i.  (16) 

Again,  as  u  changes  from  zero  towards  the  negative  side, 
cosh  u  is  positive  and  increases  from  unity  to  infinity,  but 
sinh  u  is  negative  and  increases  numerically  from  zero  to  a 
negative  infinite,  and  tanh  u  is  also  negative  and  increases 
numerically  from  zero  to  negative  unity ;  hence 

cosh  (— 00)  =  00  ,   sinh  (— 00)  =— 00  ,    tanh  (— 00  )  =— i.  (17) 

For  intermediate  values  of  u  the  numerical  values  of  these 
functions  can  be  found  from  the  formulas  of  Arts.  16,  17,  and 
are  tabulated  at  the  end  of  this  chapter.  A  general  idea  of 
their  manner  of  variation  can  be  obtained  from  the  curves  in 
Art.  25,  in  which  the  sectorial  measure  u  is  represented  by  the 
abscissa,  and  the  values  of  the  functions  cosh  u,  sinh  u,  etc., 
are  represented  by  the  ordinate. 

The  relations  between  the  functions  of  —  u  and  of  u  are 
evident  from  the  definitions,  as  indicated  above,  and  in  Art.  8. 
Thus 

cosh  (—  2/)  =  -f-  cosh  u,     sinh  (—«)=—  sinh  u,  \ 

sech  (—//)=  -|-  sech  u,     csch  {—  u)  z=  —  csch  u,    >    (18) 

tanh  (—«)=—  tanh  u^     coth  {—  u)=  —  coth  u.  ) 

Prob.  12.  Trace  the  changes  in  sech  u^  coth  u,  csch  u,  as  u  passes 
from  —  00  to  +  °^  •  Show  that  sinh  u^  cosh  u  are  infinites  of  the 
same  order  when  u  is  infinite.  (It  will  appear  in  Art.  17  that  sinh 
u,  cosh  u  are  infinites  of  an  order  infinitely  higher  than  the  order 
of  u.) 

Prob.  13.  Applying  eq.  (12)  to  figure,  page  114,  prove  tanh  u,  = 
tan  A  OP, 


116  HYPERBOLIC    FUNCTIONS.  [ChAP.  IV. 

Art.  10.    Anti-hyperbolic  Functions. 
The  equations  -  =  cosh  u,   -y  =  sinh  «,  -r  =  tanh  «,  etc.^ 
may  also  be  expressed  by  the  inverse  notation  u  =  cosh~^  -, 

y  t 

u  =  sinh~^T»  2^  =  tanh "^-7,  etc.,  which    may  be   read:    "«   is 
o  0 

the  sectorial  measure  whose  hyperbolic  cosine  is  the  ratio  x  to 
^,"  etc. ;  or  "  u  is  the  anti-h-cosine  of  x/a,''  etc. 

Since  there  are  two  values  of  u,  with  opposite  signs,  that 
correspond  to  a  given  value  of  cosh  u^  it  follows  that  if  u  be 
determined  from  the  equation  cosh  u  =  m,  where  m  is  a  given 
number  greater  than  unity,  u  is  a.  two-valued  function  of  m. 
The  symbol  cosh^^  7n  will  be  used  to  denote  the  positive  value 
of  2i.  that  satisfies  the  equation  cosh  u  =  m.  Similarly  the 
symbol  sech"^  m  will  stand  for  the  positive  value  of  u  that 
satisfies  the  equation  sech  u  =  m.  The  signs  of  the  other 
functions  sinh"^;«,  tanh'^m,  coth"'' ;;/,  csch"^ ;;/,  are  the  same 
as  the  sign  of  m.  Hence  all  of  the  anti-hyperbolic  functions 
of  real  numbers  are  one-valued. 

Prob.  14.  Prove  the  following  relations  : 

cosh"^  w  =  sinh"^  Vm^  —  i,     sinh~^w  =  ±  cosh"^  i^m^  -f-  i, 
the  upper  or  lower  sign  being  used  according  as  m  is  positive  or 
negative.     Modify  these  relations  for  sin  '^ ,  cos~^ . 

Prob.  15.  In  figure,  Art.  i,\et  OA  =  2,  O^  =  i,AOB  =  60°;  find 
the  area  of  the  hyperbolic  sector  A  OP,  and  of  the  segment  AMP, 
if  the  abscissa  of  P  is  3.     (Find  cosh"^  from  the  tables  for  cosh.) 

Art.  11.    Functions  of  Sums  and  Differences. 

(a)  To  prove  the  difference-formulas 

sinh  {u  —  v)  =  sinh  u  cosh  v  —  cosh  u  sinh  v^ 
cosh  {u  —  v)  ^  cosh  u  cosh  v  —  sinh  u  sinh 

Let  OA  be  any  radius  of  a  hyperbola,  and  let  the  sectors  AOPy 
AOQ  have  the  measures  u,  v,  then  u  —  v  is  the  measure  of  the 
sector  QOP.  Let  OB,  OQ  be  the  radii  conjugate  to  OA,  OQ; 
and  let  the  coordinates  of  P,  Q,  Q  be  (^,  ,^,),  {x,y),  (x' y  y') 
with  reference  to  the  axes  OA,  0B\  then 


^'\     (.9) 


Art.  11.] 


FUNCTIONS  OF  SUMS  AND  DIFFERENCES. 


117 


•  u  /          X         .  ,    sector  QOP      triangle  QOP    . , 
sinh  («  —  z/)  =  sinh --^ —  = °_^ —     [Art.  5. 

_  i"(-^J^i—  ^j)  sin  CO  _y^x       y  x^ 
\aj)^  sin  00        ~  b,a^      b^  a^ . 

=  sink  u  cosh  v  —  cosh  u  sinh  z; ; 


u  /  \  u  sector  06^/^      triangle  POQ'  ^  ^ 

cosh  {u  —  v)  =  cosh — F —  = ^^^ —  [Art.  5. 

K.  K 

—  K^y~^i^O  sin  c»_  y  ^, 


\aj}^  sin  07 


^1  ^, 


but 


>1 


b: 


(20) 


since  Q,  Q'  are  extremities  of  conjugate  radii ;  hence 

cosh  («  —  z;)  =  cosh  u  cosh  z/  —  sinh  u  sinh  z/. 

In  the  figures  u  is  positive  and  v  is  positive  or  negative. 
Other  figures  may  be  drawn  with  u  negative,  and  the  language 
in  the  text  will  apply  to  all.  In  the  case  of  elliptic  sectors, 
similar  figures  may  be  drawn,  and  the  same  language  will  apply, 
except  that  the  second  equation  of  (20)  will  be  x' /a^  =  —y/b^^ 
therefore 

sin  {u  —  v)  =1  sin  u  cos  v  —  cos  u  sin  v, 

cos  {u  —  v)  =  cos  ti  cos  z;  +  sin  u  sin  v, 

(b)  To  prove  the  sum-formulas 

sinh  (u-\-  v)  =  sinh  u  cosh  v  +  cosh  u  sinh  v, 
cosh  {u-\-  v)  =  cosh  u  cosh  v  +  sinh  u  sinh  z^. 

These  equations  follow  from  (19)  by  changing  v  into  —  z/, 


(21) 


118  HYPERBOLIC    FUNCTIONS.  [Chap.  IV. 

and  then  for  sinh(— z^),  cosh(— z/),  writing  —  sinh  z^,  cosh  z^ 
(Art.  9,  eqs.  (i8)). 

/  \  T-  4.1,  4.  *.     u  /      I     \       t^"h  u  ±  tanh  V  ,. 

(c)  To  prove  that  tanh  {u  ±_v)  — -—  .  (22) 

^  ^         ^  \    -^    ^       I  ±  tanh?/ tanh  z/  ^     ^ 

Writing  tanh  (2^  ±  z/)  =  -4 -,  expanding  and  dividing 

cosh  \ti  ^  V) 

numerator  and  denominator  by  cosh  u  cosh  v^  eq.  (22)  is  ob- 
tained. 

Prob.  16.  Given  cosh  «  =  2,  cosh  z'  =  3,  find  cosh  {u  -\-  v), 
Prob.  17.  Prove  the  following  identities: 

1.  sinh  27/  =  2  sinh  u  cosh  u. 

2.  cosh  2u  =  cosh^T/  +  sinh'^u  =1-1-2  sinh'  u  =  2  cosh'  u  —  i» 

3.  I  +  cosh  u  =  2  cosh'  ^u,     cosh  u  —  i  :=  2  sinh'  ^«. 

,    ,  sinh  2/  cosh  u  —  1        /cosh  «/  —  i\* 

4.  tanh  t^^  =  — . —  = r-r = : —  . 

I  -|-  cosh  u  sinh  u  \cosh  u  -{-  ij 

.   .  2  tanh  u  .  T  4-  tanh'  u 

5.  sinh  2U  =  --^— ,     cosh  2U  — -^— . 

*'  I  —  tanh   u  i   —  tanh  u 

6.  sinh  32/  =  3  sinh  «  +  4  sinh''  «,  cosh  32/  =  4  cosh' «  —  3  cosh  2^. 

,        ,     •   ,  I  +  tanh  ^7/ 

7.  cosh  u  4-  sinh  «  = — ;— i"  • 

'  I  —  tanh  -^u 

8.  (cosh  «  -\-  sinh  «)(cosh  z;  +  sinh  z^)=cosh  {u -\- v)  -\-  sinh  («  -\-  v). 

9.  Generalize  (8);  and  show  also  what  it  becomes  when  u=v=  .  . . 

10.  sinh'jt:  cos'^  +  cosh'^  sin'^*  =  sinh'^  -f~  sin'j. 

11.  cosh'^m  ±  cosh"^;?  =  cosh~^L^^^  i   V  (w'  —  i)(«'  — i)J. 

12.  sinh~^;«  ±  sinh"^;?  =  sinh~^[w y  i  -\-  n^  ±  nyi  -\-  w'J. 

Prob.  18.  What  modifications  of  signs  are  required  in  (21),  (22), 
in  order  to  pass  to  circular  functions  ? 

Prob.  19.  Modify  the  identities  of  Prob.  17  for  the  same  purpose. 

Art.  12.    Conversion  Formulas. 
To  prove  that 

cosh  «,+  cosh  «,  =  2  cosh  i{u^-\-  u,)  cosh  i(«,"-  u^, 
cosh  «,—  cosh  ^,  =  2  sinh  i{u^  -\- u^ sinh  i{u^—  u,\ 
sinh  u,  +  sinh  u^  —  2  sinh  ^(2/,  +  u^  cosh  \{u^—  u,),   1 
sinh  u^  —  sinh  «,  =  2  cosh  i(«,  +  «,)  sinh  i(^i—  ^O-  J 


Art.  13.]  limiting  ratios.  119 

From  the  addition  formulas  it  follows  that 

cosh  {u  -{-  v)-\-  cosh  {u  —  v)  =  2  cosh  u  cosh  Vj 

cosh  {u  -\-v)  —  cosh  (u  —  v)  =  2  sinh  u  sinh  Vy 

sinh  {u  -\-  v)-{-  sinh  {u  —  v)  =  2  sinh  «  cosh  v, 

sinh  (?/  -}-  z/)  —  sinh  (/^  —  z/)  =  2  cosh  u  sinh  ^', 

and  then  by  writing  u  -{-v'=  u^  ,    u  —  v  —  u^,    u  =  ^{u^  +^9). 
V  =  ^{u^  —  u^),     these  equations  take  the  form  required. 

Prob.  20.  In  passing  to  circular  functions,  show  that  the  only 
modification  to  be  made  in  the  conversion  formulas  is  in  the  alge- 
braic sign  of  the  right-hand  member  of  the  second  formula. 

_,     ,  -,.      ...    cosh  2U  -\-  cosh  4V  cosh  2U  -\-  cosh  4V 

Prob.  21.  Simphfy  -^-r- ■ — .   ,   ^  ,  r t~-' 

smh  2U  -\-  sinh  4v  cosh  2U  —  cosh  4V 

Prob.  22.  Prove  sinh'jc  —  sinh'j'  =  sinh  {x  -{-y)  sinh  (x  —y)» 
Prob.  23.  Simplify  cosh^jc  cosh'j^  ±  sinh^jc  sinh'^j. 
Prob.  24.  Simplify  cosh'jv  cos'*  y  -f  sinh':^;  sin'^. 

Art.  13.    Limiting  Ratios. 

To  find  the  limit,  as  u  approaches  zero,  of 

sinh  u  tanh  u 

u  u 

which  are  then  indeterminate  in  form. 

By  eq.  (14),  sinh  uy  u>  tanh  u  ;  and  if  sinh  u  and  tanh  « 
be  successively  divided  by  each  term  of  these  inequalities,  it 
follows  that 

sinh  u  . 

I  <  <  cosh  Uf 


.         ^  tanh  u    ^ 
sech  u  <  <  I.' 

u 


but  when  «i^o,  cosh  u  =  i^  sech  u—  i,  nence 

Hm.  sinh^^  ^  lim.   t^nhu   ^  ^  ^^4) 


120 


HYPERBOLIC    FUNCTIONS. 


rCHAP.  IV. 


Art.  14.    Derivatives  of  Hyperbolic  Functions. 
To  prove  that 

df(sinh  u) 


du 

^(cosh  ti) 
du 

^(tanh  li) 
du 

^(sech  u) 
du 

d(Q.o\h  u) 

du 
^(csch  ?/) 


=  cosh  Uf 

=  sinh  u, 

=  sech'  u, 

=  —  sech  u  tanh  u, 

—  —  csch^  u, 

=  —  csch  u  coth  u. 


du 

(a)  Let  y  =  sinh  z^, 

Jj|/  =  sinh  (u  -\-  ^u)  —  sinh  u 

=  2  cosh  1(22/  +  z/2/)  sinh  ^z/«, 

Jjj/                                  sinh  J J^/  • 
-f-  =  cosh  (2/  -f-  i^^j  — T-A • 

Take  the  limit  of  both  sides,  as  Ju  =  o,  and  put 

jy       df       ^(sinh  u) 
Ju       du  ~        du 
lim.  cosh  {u  -\-  ^Au)  =  cosh  u, 
sinh  ^Au 


lim.        ,    . 

^Au 

^(sinh  u)  _ 
du 


—  I ;         (see  Art.  13) 


cosh  u. 


(J?)  Similar  to  (a). 

^(tanh  u)        d    sinh  u 


w 


du 


du '  cosh  u 
cosh'  2/  —  sinh'  u 
cosh'  // 


(25) 


= rr^  =  sech'  u, 

cosh  ;/ 


Art.  15,]         derivatives  of  hyperbolic  functions.  121 

{d)     Similar  to  {c), 

.  .      ^(sech  u)        d  I  sinh  u 

(e)     J =  -J- . 7^  =  —  — ri—  =  --  seen  u  tanh  u. 

^  ^  dii  du    cosh  u  cosh  u 

(/)     Similar  to  {e). 

It  thus  appears  that  the  functions  sinh  u^  cosh  u  reproduce 
themselves  in  two  differentiations ;  and,  similarly,  that  the 
circular  functions  sin  u,  cos  u  produce  their  opposites  in  two 
differentiations.  In  this  connection  it  may  be  noted  that  the 
frequent  appearance  of  the  hyperbolic  (and  circular)  functions 
in  the  solution  of  physical  problems  is  chiefly  due  to  the  fact 
that  they  answer  the  question :  What  function  has  its  second 
derivative  equal  to  a  positive  (or  negative)  constant  multiple 
of  the  function  itself?  (See  Probs.  28-30.)  An  answer  such  as 
y  z=  cosh  mx  is  not,  however,  to  be  understood  as  asserting  that 
mx  is  an  actual  sectorial  measure  and  y  its  characteristic  ratio  ; 
but  only  that  the  relation  between  the  numbers  mx  and  y  is  the 
same  as  the  known  relation  between  the  measure  of  a  hyper- 
bolic sector  and  its  characteristic  ratio  ;  and  that  the  numerical 
value  of  ^  could  be  found  from  a  table  of  hyperbolic  cosines. 

Prob.  25.  Show  that  for  circular  functions  the  only  modifica- 
tions required  are  in  the  algebraic  signs  of  (d),  (d). 

Prob.  26.  Show  from  their  derivatives  which  of  the  hyperbolic 
and  circular  functions  diminish  as  u  increases. 

Prob.  27.  Find  the  derivative  of  tanh  u  independently  of  the 
derivatives  of  sinh  «,  cosh  u. 

Prob.  28.  Eliminate  the  constants  by  differentiation  from  the 
equation^  =  A  cosh  mx  +  B  sinh  mx,  and  prove  that  d^/dx^  =  my. 

Prob.  29.  Eliminate  the  constants  from  the  equation 

y  =  A  cos  mx  -\-  B  sin  mx^ 

and  prove  that  d  "^y/dx'  =  —  my. 

Prob.  30.  Write  down  the  most  general  solutions  of  the  differen- 
tial equations 

dy       ,      dy  ,      dy 


133  HYPERBOLIC    FUNCTIONS.  TChap.  IV'.. 

Art.  15.    Derivatives  of  Anti-hyperbolic  Functions.. 
{a) 

ic) 
id) 

W 
(/) 
(a)  Let     u  =  sinh"'  x,     then  x  =  sinh  «,  dx  =  cosh  u  du 


^(sinh"' 

x)_          I 

dx 
4cosh- 

Vx'  +  I 

dx 
4tanh-' 

dx 
4coth-^ 

I    -  X'jx<^' 

x)_           I    n 

dx 
d{scch-' 

dx 
4csch-^ 

dx 

x\/x'  -\-l 

(26). 


=  Vi  +  sinh'  2/  ^/^  =  4/1  +  ;r''  du,     du  =  dx/  Vi  -f  x\ 
{b)  Similar  to  {a), 
{c)  Let     u  =  tanh"'  x,     then  ;?r  =  tanh  u,  dx  =  sech'  u  du 

=  (l  —  tanh'  ^y?^  =  (i  —  ^V^»     ^«  =  ^Vi  —  ^'• 
(^)  Similar  to  (^). 

^(5££ii::^)^/Ush--i)=.=i/(i-iy=  — ^L=- 

^  ^  ^;ir  dx\  xl       x^  f   \x^        I       xVl—  x 

(/)  Similar  to  (e). 


Prob.   31.  Prove 

d{s\n~^  x)  _ 
dx 


Vi  -  x^' 


^(cos-*^)  _ 


dx 


Vl 


X 


^(tan~^  x)  __       I 

dx        ~  I  +  -^^ 


^(cot~^  x)  _  I 

dx        "       I  -f-  •^'* 


Art.  16.]  expansion  of  hyperbolic  functions.  133- 

Prob.  32.  Prove  ^  ^ 

^sinh"  -  =  — 7^==,         ^cosh    -  = 


,     ,  X  adx    "1  ,       ,     ,  -^  (^dx    ~\ 

d  tanh"   -  =  -. =         ,     ^coth"  -  = v— — 2 

a       a   —  X  Jx<a  a  x   —  a  Jx>m, 

Prob.  33.  Find  ^(sech~^  x)  independently  of  cosh"^  x. 

Prob.  34.  When  tanh~*  x  is  real,  prove  that  coth  '  x  is  imagi- 
nary, and  conversely;  except  when  x  =  1. 

^     ,  sinh"'  X      cosh"^  x         . 

Prob.  %^.  Evaluate      —, ,     -1 ,     when  a:  =  x  . 

^^  log  X    ^        log  Jf    ' 

Art.  16.    Expansion  of  Hyperbolic  Functions. 
For  this  purpose  take  Maclaurin's  Theorem, 

/in)  =  Ao)  +  «/'(o)  +  J,  «y"(o)  +  ^  «y"'(o)  + . . ., 

and  put  f{u)  =  sinh  Uy     /\u)  =  cosh  u,    f''{u)  =  sinh  u,  .  .  ., 
then         /{o)  =  sinh  0  =  0,    /\o)  =  cosh  0=1,...; 

hence  sinh  u  =  u +—^u'  -^ —^u' -{-  .  ,  .;  (27) 

and  similarly,  or  by  differentiation, 

cosh  U  =    I   -\ r/^'H :«*  +  ....  (28) 

2!  4! 

By  means  of  these  series  the  numerical  values  of  sinh  u, 
cosh  u,  can  be  computed  and  tabulated  for  successive  values  of 
the  independent  variable  u.  They  are  convergent  for  all  values 
of  u,  because  the  ratio  of  the  nth  term  to  the  preceding  is  in 
the  first  case  u^/{2n  —  \){2n  —  2),  and  in  the  second  case 
»y(2«  —  2)(2«  —  3),  both  of  which  ratios  can  be  made  less  than 
unity  by  taking  n  large  enough,  no  matter  what  value  u  may 
have. 


(29) 


124  HYPERBOLIC    FUNCTIONS.  [ChAP.  IV. 

From  these  series  the  following  can  be  obtained  by  division  : 

tanh  u  =  u-\u'  +  -f^u'  -j-  ^VV'^'  +  •  •  • ,  ^ 
sech  u=zi  —  :^u'-{-  -f^u"  —  -^\u'  +  .  .  . , 
u  coth  u  =  I  -\-iu^  —  -^-^U*  +  -fff 3«'—  •  .  . , 
u  csch  u  =i—iu'  +  -^^-^u* -  tAW^'+ 

These  four  developments  are  seldom  used,  as  there  is  no 
observable  law  in  the  coefficients,  and  as  the  functions  tanh  u, 
sech  Uy  coth  u,  csch  u,  can  be  found  directly  from  the  previously 
computed  values  of  cosh  u,  sinh  u. 

Prob.  36.  Show  that  these  six  developments  can  be  adapted  to 
the  circular  functions  by  changing  the  alternate  signs. 

Art.  17.     Exponential  Expressions. 
Adding  and  subtracting  (27),  (28)  give  the  identities 

cosh  u  +  sinh  u  =  i  4- u  4-  -p«'  -I — -u^  -\ — -u*  -j-  .  .  .  =  ^«, 

2!  3!  4! 

cosh  u  —  sinh  u  —  i  —  u-\ -u^ -u^  -\-  —-u*  —  .  .  .  =  e'"*, 

2!  3'  4! 

hence    cosh  u  —  ^{f  +  ^~'*),     sinh  u  =  i{e'*  —  e~'*),       1 

^«  _  ^  -«  2  r     (30) 

tanh  u  =  — ; ,         sech  u  = ,     etc.  1 

^"^^-^  ^'«+^-"  J 

The  analogous  exponential  expressions  for  sin  u,  cos  u  are 
cos  u  =  ^-(^"^  +e~'*'\     sin  u  =  — .(^'**  —  ^-«'),      (t  =  ^  —  i) 

2  21 

where  the  symbol  ^"*  stands  for  the  result  of  substituting  ui  for 
X  in  the  exponential  development 

This  will  be  more  fully  explained  in  treating  of  complex 
numbers,  Arts.  28,  29. 


Art.  18.]  EXPANSION    OF    ANTI-FUNCTIONS.  125 

Prob.  37.  Show  that  the  properties  of  the  hyperbolic  functions 
could  be  placed  on  a  purely  algebraic  basis  by  starting  with  equa- 
tions (30)  as  their  definitions  ;  for  example,  verify  the  identities : 

sinh  {—  u)  =  —  sinh  u,     cosh  (—  u)  =  cosh  u, 

cosh'  u  —  sinh^  u—i,     sinh  (u-]-v)  =  sinh  u  cosh  v  +  cosh  u  sinh  Vf 

^'^(cosh  mu)  „        ,  ^^(sinh  ?nu)  ,    .   , 

— ^^ — — r =  m  cosh  mu,     —z —  nr  sinh  mu, 

du  du^ 

Prob.  38.  Prove  (cosh  u  +  sinh  uY  =  cosh  nu  -}-  sinh  nu. 

Prob.  39.  Assuming  from  Art.  14  that  cosh  «,  sinh  u  satisfy  the 
differential  equation  dy/du^  =y,  whose  general  solution  may  be 
written  y  =  Ae**  +  Be'"",  where  ^,  B  are  arbitrary  constants  ;  show 
how  to  determine  A,  B  in  order  to  derive  the  expressions  for  cosh  u^ 
sinh  u,  respectively.     [Use  eq.  (15).] 

Prob.  40.  Show  how  to  construct  a  table  of  exponential  func- 
tions from  a  table  of  hyperbolic  sines  and  cosines,  and  vice  versa. 

Prob.  41.  Prove  u  =  log<,  (cosh  u  -\-  sinh  u). 

Prob.  42.  Show  that  the  area  of  any  hyperbolic  sector  is  infinite 
when  its  terminal  line  is  one  of  the  asymptotes. 

Prob.  43.  From  the  relation  2  cosh  u  —  e""  -\-  e'""  prove 

2**"Xcosh  2/)'*=cosh  nu  +  n  cosh  («— 2)«  +  i«(«— i)  cosh  (n—4)u  +  . .  .> 

and  examine  the  last  term  when  n  is  odd  or  even. 

Find  also  the  corresponding  expression  for  2""^  (sinh  «)'*. 

Art.  18.    Expansion  of  Anti-Functions. 
Since      4sinh-  .)  ^        i_  ^ 

2  24  246 

hence,  by  integration, 

23*245         2467 

the   integration-constant   being   zero,    since    sinh"'  x  vanishes 
with  X.     This  series  is  convergent,  and  can  be  used  in  compu- 


126  HYPERBOLIC    FUNCTIONS.  [Chap.  IV. 

tation,  only  when  x  <  i.  Another  series,  convergent  when 
;i;  >  I,  is  obtained  by  writing  the  above  derivative  in  the  form 

ax  x\        X I 

xL  2;tr'^2  4;i:^        2  A,  6  x'^         S 

.-.  sinh-^  X  =  C+\og  x-\ —  ^  ~4+-  -  i  r-e-'  •  •  >     (32) 

'  2  2;ir'      2  4  4;ir*  '  2  4  6  6;ir«  ^^  ^ 

where  C  is  the  integration-constant,  which  will  be  shown  in 
Art.  19  to  be  equal  to  log^  2. 

A  development  of  similar  form  is  obtained  for  cosh~*  ;tr;  for 

c;lx  x\        XI 

=:iri4-il  +  i3±  ,   I  3  5  i   ,        1 

xL     ^  2  X'^2  4X*^2  46  X'^"'J 

hence 

^   ,    t  II  131  I35I  /x 

<osh-.  =  C+log^---,--^-.--J|g^.-...,    (33) 

in  which  C  is  again  equal  to  log,  2  [Art.  19,  Prob.  46].  In 
order  that  the  function  cosh"^;i;  maybe  real,  ;r  must  not  be 
less  than  unity ;  but  when  x  exceeds  unity,  this  series  is  con- 
vergent, hence  it  is  always  available  for  computation. 

Again,     ^^-l^  =  -L-,=  i+^^  +  .'^. '+..., 
dx  I       X 

and  hence  tanh"'  x  —  x -\- - x"  -\;- -x^ -\- -x'  -\' ,  ,  ,,     (34) 

From  (32),  (33),  (34)  are  derived : 
I 

X 

^  2.2      2.4.4      2.4.6.6 


Art.  19  ]    logarithmic  expression  of  anti-functions.  127 

•  111       I       11,131        i3c;i, 
X      X      2  IX       2    ^'^x       2  46  yx' 

=   C_10g.  +  _-_l_+_A_l__...;     (36) 

coth-  .  =  tanh-  ^  =  i  +  3-^.  +  5-i.  +  7-^  +  •  •  ••  (37) 

Prob.  44.  Show  that  the  series  for  tanh"^  x^  coth"^  x^  sech~^  x^ 
are  always  available  for  computation. 

Prob.  45.  Show  that  one  or  other  of  the  two  developments  of  the 
inverse  hyperbolic  cosecant  is  available. 

Art.  19.    Logarithmic  Expression  of  Anti-Functions. 


Let  X  =  cosh  u,      then  ^ x"  —  i  =  sinh  u\ 


therefore  x  -\-  Vx*  —  i  =  cosh  u  -\-  sinh  u  =  e^^ 

and  //,  =  cosh"^;r,  =  log(;r  +  1^-^' —  0'  (3^) 

Similarly,    sinh~^;r  =  log  (^  +  Vx*  +  l)-  (39) 


Also  sech"^;r=  cosh~^-  =  log  -^ ,  (40) 

X  X 


csch~*;ir  =  sinh"*-  =  log  -—^ — — .  (41) 

X  X 

Again,  let  x  =  tanh  u  = 


therefore  =  —  =  ^,  ' 

I  —  X      e 

2u  =  log      *"  ^,     tanh"*;i;=J  log     "^    ;        (42) 
I  —  X  I  —  X 

and  coth-*;r  =  tanh"*-  =  i  log  ^^ .  (43) 

Prob.  46.  Show  from  (39),  (40),  that,  when  :«:-i  00, 

sinh~*;if  —  log  X  -^  log  2,  cosh"  ^x  -  log  x  -i  log  2, 

and  hence  show  that  the  integration-constants  in  (32),  (33)  are  each 
-equal  to  log  2. 


128  HYPERBOLIC    FUNCTIONS.  [Chap.  IV. 

Prob.  47.  Derive  from  (42)  the  series  for  tanh  '^x  given  in  (34). 
Prob.  48.  Prove  the  identities: 

jc — I  x^ — I 

los^=2 tanh "^—-—= tanh '^  - — =sinh~^i(:v— ^-i)=cosh"H(A:+^-^); 

°  x+i  X  +1  ^  ^^ 

log  sec  X  =  2  tanh"^  tan''  ^x;  log  esc  x=  2  tanh'  '  tanXi^r  +  ix); 
log  tan  X  =  ~~  tanh"^  cos  2x:  =  —  sinh"^  cot  2x  =  cosh~^  esc  2X. 

Art.  20.  The  Gudermanian  Function. 

The  correspondence  of  sectors  of  the  same  species  was  dis- 
cussed in  Arts.  1-4.  It  is  now  convenient  to  treat  of  the 
correspondence  that  may  exist  between  sectors  of  different 
species. 

Two  points  P^,  P^ ,  on  any  hyperbola  and  ellipse,  are  said  to 
correspond  with  reference  to  two  pairs  of  conjugates  O^A^ , 
O^B^ ,  and  O^A^,  O^B^,  respectively,  when 

•^i/«,  =  ajx^,  (44) 

and  when  j/i,jK2  have  the  same  sign.  The  sectors  A^O.P^, 
A^O^P^  are  then  also  said  to  correspond.  Thus  corresponding 
sectors  of  central  conies  of  different  species  are  of  the  same 
sign  and  have  their  primary  characteristic  ratios  reciprocal. 
Hence  there  is  a  fixed  functional  relation  between  their  re- 
spective measures.  The  elliptic  sectorial  measure  is  called 
the  gudermanian  of  the  corresponding  hyperbolic  sectorial 
measure,  and  the  latter  the  anti-gudermanian  of  the  former. 
This  relation  is  expressed  by 

SJK,  =  gd  SJK, 
or     ^^  =  gd  u,     and     u  =  gd~'^v.  (45) 

Art.  21.    Circular  Functions  of  Gudermanian. 

The  six  hyperbolic  functions  of  u  are  expressible  in  terms 
of  the  six  circular  functions  of  its  gudermanian  ;  for  since 

X  X 

—  —  cosh  u,       —  =  cos  Vy  (see  Arts.  6,  7) 

in  which  u,  v  are  the  measures  of  corresponding  hyperbolic 
and  elliptic  sectors, 


Art.  22.]  gudermanian  angle.  129' 

hence  cosh  u  =  sec  v,  [eq.  (44)] 


(46) 


sinh  u  =  r  secV  —  i  =  tan  v, 

tanh  u  =  tan  ^/sec  v  =  sin  v, 

coth  u  =  esc  V, 

sech  li  =  cosz^, 

csch  u  =  cot  z/. 

The  gudermanian  is  sometimes  useful  in  computation ;  for 
instance,  if  sinh  u  be  given,  v  can  be  found  from  a  table  of 
natural  tangents,  and  the  other  circular  functions  of  v  will  give 
the  remaining  hyperbolic  functions  of  u.  Other  uses  of  this 
function  are  given  in  Arts.  22-26,  32-36. 

Prob.  49.  Prove  that  gdu  =  sec"^(cosh  u)  =  tan~^(sinh  u) 

=  cos~^(sech  u)  =  sin"*(tanh  u)^ 

Prob.  50.  Prove       gd~*z;  =  cosh"^(sec  v)  =  sinh~^  (tan  v) 

=  sech" ^ (cos  v)  =  tanh~^(sin  v), 

Prob.  51.   Prove      gd  o  =  o,  gd  00  =  ^7r,       gd(—  00)  =  —  -J^r^ 

gd-^  0=0,  gd- '(iTt)  =cx),  gd-i(— j7r)  =  -oo. 

Prob  52.  Show  that  gd  u  and  gd"  ^  v  are  odd  functions  of  u,  v. 

Prob.  53.  From  the  first  identity  in  4,  Prob.  19,  derive  the  rela- 
tion tanh  ^u  =  tan  iv. 

Prob.  54.  Prove 

tanh"  ^(tan  «)  =  ^  gd  2«,  and  tan"^(tanh  x)  =  i  gd"^2.;if. 

Art.  22.     Gudermanian  Angle 

If  a  circle  be  used  instead  of  the  ellipse  of  Art.  20,  the 
gudermanian  of  the  hyperbolic  sectorial  measure  will  be  equal 
to  the  radian  measure  of  the  angle  of  the  corresponding  circular 
sector  (see  eq.  (6),  and  Art.  2,  Prob.  2).  This  angle  will  be 
called  the  gudermanian  angle  ;  but  the  gudermanian  function  Vy 
as  above  defined,  is  merely  a  number,  or  ratio  ;  and  this  number 
is  equal  to  the  radian  measure  of  the  gudermanian  angle  0, 
which  is  itself  usually  tabulated  in  degree  measure ;  thus 

6  =  iSo°  v/tt (47> 


130 


HYPERBOLIC    FUNCTIONS. 


[Chap.  IV. 


Prob.  55.  Show  that  the  gudermanian  angle  of  u  may  be  construct- 
ed as  follows: 

Take  the  principal  radius  OA  of  an  equilateral  hyperbola,  as  the 
initial  line,  and  OP  as  the  terminal 
line,  of  the  sector  whose  measure  is  u\ 
from  M^  the  foot  of  the  ordinate  of 
P^  draw  MT  tangent  to  the  circle 
whose  diameter  is  the  transverse  axis; 
then  AOTy^  the  angle  required.* 

Prob.  56.     Show   that   the   angle    B 
never  exceeds  90°. 

Prob.  57.  The  bisector  of  angle  AOT 

bisects  the  sector  AOP  (see  Prob.  13, 

Art.  9,  and  Prob.  53,  Art.  21),  and  the  line  AP.     (See  Prob.  i,  Art.  3.) 

Prob.  58.  This  bisector  is  parallel  to  TPy  and  the  points  T,P 

are  in  line  with  the  point  diametrically  opposite  to  A. 

Prob.  59.   The  tangent   at  P  passes  through  the   foot   of   the 
ordinate  of  T^  and  intersects  TM  on  the  tangent  at  A, 

Prob.  60.  The  angle  APM  is  half  the  gudermanian  angle. 

Art.  23.    Derivatives  of  Gudermanian  and  Inverse. 
Let  V  ■=  %^  Uy    u  =.  gd"*  Vf 

then  sec  7)  =  cosh  u, 

sec  V  tan  vdv  =  sinh  u  du, 
sec  vdv  =  du, 
therefore  <^(gd-'  v)  =  sec  vdv,  (48) 

Again,  dv  =  cos  v  du    =  sech  u  du, 

therefore  ^(gd  u)  =  sech  u  du.  (49) 

Prob.  61.  Differentiate: 


y  =  sinh  u  —  gd  u, 


y  =  sin  V  +  gd~^  v, 


tanh  u  sech  u  -{-  gd  u^     y  =  tan  7^  sec  v  -\-  gd' 


V. 


*  This  angle  was  called  by  Gudermann  the  longitude  of  u,  and  denoted  by  lu. 
His  inverse  symbol  was  |t;  thus  u  =  '^{lu).  (Crelle's  Journal,  vol.  6,  1830.) 
Lambert,  who  introduced  the  angle  6,  named  it  the  transcendent  angle.  (Hist, 
de  I'acad,  roy.  de  Berlin,  1761).  Hottel  (Nouvelles  Annales,  vol.  3,  1864) 
called  it  the  hyperbolic  amplitude  of  u,  and  wrote  it  amh  u,  in  analogy  with  the 
amplitude  of  an  elliptic  function,  as  shown  in  Prob.  62.  Cayley  (Elliptic 
Functions,  1876)  made  the  usage  uniform  by  attaching  to  the  angle  the  name 
of  the  mathematician  who  had  used  it  extensively  in  tabulation  and  in  the 
theory  of  elliptic  functions  of  modulus  unityo 


Art.  24.]       series  for  gudermanian  and  its  inverse.  131 

Prob.  62.  Writing   the   "elliptic  integral  of  the  first  kind"  in 
the  form 


J       Vi  -  k'  sin' 


0' 

K  being  called  the  modulus,  and  0  the  amplitude;  that  is, 

0  =  am  u,  (mod.  /c), 

show  that,  in  the  special  case  when  /c  =  i, 

u  =  gd~^  0,  am  2^  =  gd  u,     sin  am  «  =  tanh  «, 

cos  am  /^  =  sech  u,       tan  am  w  =  sinh  u; 

and  that  thus  the  elliptic  functions  sin  am  u,  etc.,  degenerate  into 
the  hyperbolic  functions,  when  the  modulus  is  unity.* 

Art.  24.    Series  for  Gudermanian  and  its  Inverse. 

Substitute  for  sech  u,  sec  v  in  (49),  (48)  their  expansions, 
Art.  16,  and  integrate,  then 

gdu  =  u-^u'  +  -^u"  -  ^%)^u'  +  . . .  (50) 

gd-,;  =  ^  +  iz.'  +  i^v^  +^Hir^^  +  .  . .  (51) 

No  constants  of  integration  appear,  since  gd  u  vanishes  with 
«,  ana  gd''^*  with  v.  These  series  are  seldom  used  in  compu- 
tation, as  gd  u  is  best  found  and  tabulated  by  means  of  tables 
of  natural  tangents  and  hyperbolic  sines,  from  the  equation 

gd  «  =  tan~*(sinh  ti)^ 
and  a  table  of  the  direct  function  can  be  used  to  furnish  the 
numerical  values  of  the  inverse  function  ;  or  the  latter  can  be 
obtained  from  the  equation, 

gd"V  =  sinh"*(tan  v)  =  cosh~'(sec  v). 
To  obtain  a  logarithmic  expression  for  gd"V,  let 
gd"'z^  =  u,  V  =  gdu, 

*  The  relation  gd  u  =  am  u,  (mod.  i),  led  HoUel  to  name  the  function  gd  u, 
the  hyperbolic  amplitude  of  u,  and  to  write  itamh  u  (see  note,  Art.  22).  In  this 
connection  Cayley  expressed  the  functions  tanh  u,  sech  u,  sinh  u  in  the  form 
sin  gd  u,  cos  gd  u,  tan  gd  u,  and  wrote  them  sg  u,  eg  u,  tg  u,  to  correspond 
with  the  abbreviations  sn  u,  en  u,  dn  u  for  sin  am  u,  cos  am  u,  tan  am  u. 
Thus  tanh  «  =  sg  «  =  sn  «,  (mod.  i);  etc. 

It  is  well  to  note  that  neither  the  elliptic  nor  the  hyperboHc  functions 
received  their  names  on  account  of  the  relation  existing  between  them  in  a 
special  case.     (See  foot-note,  p.   107.) 


132 


HYPERBOLIC    FUNCTIONS. 


[Chap.  IV. 


therefore  sec  v  =  cosh  u,     tan  v  =  sinh  «, 

sec  V  +  tan  v  =  cosh  u  -j-  sinh  u  =  e^, 

I  -|-  sin  V      I*—  cos  {\7t  -\-  V)       ^       ,,       I    1   \ 


cos  V  sin  (^TT  -J-  2^) 

?/,  =  gd  'z;,  =  log,  tan  {Itv  +  ^^). 


(52) 


Prob.  63.   Evaluate 


gd  u 


gd  V  —  z/' 


z^  —  v~] 


Prob.  64.  Prove  that  gd  z^  —  sin  u  is  an  infinitesimal  of  the  fifth 
order,  when  u  =:  o. 

Prob.  65.  Prove  the  relations 

iTt  -{-iv=  tan"  V" ,    i7t  —  iv  —  tan"  '<?"" . 

Art.  25.  Graphs  of  Hyperbolic  Functions. 

Drawing  two  rectangular  axes,  and  laying  down  a  series  of 
points  whose  abscissas  represent,  on  any  convenient  scale,  suc- 
cessive values  of  the  sectorial  measure,  and  whose  ordinates 
represent,  preferably  on 
the  same  scale,  the  corre- 
sponding values  of  the 
function  to  be  plotted,  the 
locus  traced  out  by  this 
series  of  points  will  be  a 
graphical  representation  of 
the  variation  of  the  func- 
tion as  the  sectorial  meas- 


Art.  25.]  graphs  of  the  hyperbolic  functions.  133 

ure  varies.     The  equations  of  the  curves  in  the  ordinary  carte- 
sian notation  are ; 

Fig.  Full  Lines.  Dotted  Lines. 

A        J  =  cosh  x^        y  =  sech  x ; 

B         /  =  sinh  x^        y  z=.  csch  x  ; 

C         ^  =  tanh  X,       y  =  coth  x ; 

D        y  =  gd  X. 

Here  x  is  written  for  the  sectorial  measure  u,  SLud  y  for  the 
numerical  value  of  cosh  u,  etc.  It  is  thus  to  be  noted  that  the 
variables  Xy  y  are  numbers,  or  ratios,  and  that  the  equation 
y  =  cosh  X  merely  expresses  that  the  relation  between  the 
numbers  x  and  y  is  taken  to  be  the  same  as  the  relation  be- 
tween a  sectorial  measure  and  its  characteristic  ratio.  The 
numerical  values  of  cosh  u,  sinh  u,  tanh  u  are  given  in  the 
tables  at  the  end  of  this  chapter  for  values  of  u  between  o  and 
4.  For  greater  values  they  may  be  computed  from  the  devel- 
opments of  Art.  16. 

The  curves  exhibit  graphically  the  relations : 
sech  u  =  — -— ,      csch  u  =  -:— — ,    coth  u 


cosh  u  sinh  u  tanh  u 

cosh  u  <  I,     sech  uy>  i,     tanh  «  >  i,     gd  «  <  ^tt,  etc.; 
sinli  {—u)=—  sinh  u,     cosh  (—  u)  =  cosh  u, 
tanh  (—«)=  —  tanh  u,     gd  (—  u)    =  —  gd  u,  etc.; 
cosh  0=1,     sinh  0  =  0,     tanh  0  =  0,    csch  (o)  =00  ,  etc.; 
cosh  (±  00 )  =  00  ,  sinh  (±  00 )  =  ±00  ,  tanh  (±  00 )  =  ±  i,  etc. 

The  slope  of  the  curve  j  =  sinh  x  is  given  by  the  equation 
dy/dx  =  cosh  x,  showing  that  it  is  always  positive,  and  that 
the  curve  becomes  more  nearly  vertical  as  x  becomes  infinite. 
Its  direction  of  curvature  is  obtained  from  d'^y/dx^  =  sinh  x, 
proving  that  the  curve  is  concave  downward  when  x  is  nega- 
tive, and  upward  when  x  is  positive.  The  point  of  inflexion  is 
at  the  origin,  and  the  inflexional  tangent  bisects  the  angle 
between  the  axes. 


134 


HYPERBOLIC    FUNCTIONS. 


[Chap.  IV. 

The  direction  of  curvature  of  the  locus  jv  =  sech  x  is  given 
hy  d'y/dx'  —  sech  x{2  tanh'^r  —  i),  and  thus  the  curve  is  con- 
cave downwards  or  upwards 
according  as  2  tanh'  jr  —  i  is 
negative  or  positive.  The  in- 
flexions occur  at  the  points 
X  =i  ±  tanh-\707,  =  ±  .881, 
y  =  .707 ;  and  the  slopes  of 
the  inflexional  tangents  are 
±  1/2. , 

The  curve  y  =  csch  x  is 
asymptotic  to  both  axes,  but 
approaches  the  axis  of  x  more 
rapidly  than  it  approaches  the 
axis  of  y,  for  when  x  =  '^,  y  is 
only  .1,  but  it  is  not  till  y  =  10 
The  curves  y  =  csch  x,y  =z  sinh  x 


-1-- 


1 


that  X  is  so  small  as  .1. 
cross  at  the  points  at  =  ± 


,881,7=  ±  I. 


1«- 

-- 

- — 

-r- 

-=t- 

-=t- 

-=i- 

1 

r — ^ 

— J — 

-t— 

-i— 

— i 

ttTT- 

" 

D 

Prob.  66.  Find  the  direction  of  curvature,  the  inflexional  tan- 
gent, and  the  asymptotes  of  the  curves  j^  =  gdx,y  =  tanh  :«:. 

Prob.  67.  Show  that  there  is  no  inflexion-point  on  the  curves 
y  =  cosh  x,y  =^  coth^. 

Prob.  68.  Show  that  any  line  y  =  mx  +  n  meets  the  curve 
y  =  tanh  x  in  either  three  real  points  or  one.  Hence  prove  that 
the  equation  tanh  x  =  mx  -}-  n  has  either  three  real  roots  or  one. 
From  the  figure  give  an  approximate  solution  of  the  equation 
tanh  X  =  X  —  i. 


Art.  26.]  elementary  integrals.  135 

Prob.  69.  Solve  the  equations:    cosh  x  —  x -{-  2;  sinh  x  =  ^x; 

gd  X  =  X  —  ^Tt. 

Prob.  70.  Show  which  of  the  graphs  represent  even   functions, 
and  which  of  them  represent  odd  ones. 


Art.  26.    Elementary  Integrals. 

The  following  useful  indefinite  integrals  follow  from  Arts. 
14,15,23: 

Hyperbolic.  Circular. 

1.  /sinh  udu  =  cosh  u,  f  sm  u  du  =  —  cos  Uy  ." 

2.  /  cosh  u  du  =  sinh  u^  j  cos  u  du  =z  sin  u, 

3.  /  tanh  u  du  =^  log  cosh  u,  j  tan  udu  =  —  log  cos  Uy 

4.  /  coth  u  du  ^^  log  sinh  u,  I  cot  u  du  =1  log  sin  u, 

5.  jcsch  udu  =  log  tanh  —  ,      /  esc  u  du  =  log  tan  - , 

=  —  sinh-\csch  ?/),  =  —  cosh -'(esc  «), 

6.  /  sech  u  du  =^  gd  u,  j  sec  u  du  =  gd"'  u, 


C      dx  .   ^   ^x  ^        C       d^  .     , 

7.    /      ,  ^        -  =  Sinn-  — ,         /  — ,  ,  =  sin~ 


X 


^-  f  ./7^=^°^^-'^'      f     -^^^    =^°^ 


dx  .  ;ir 

|/;r'  -  ^'^  a'      ^     Va'  -  x'  "~     a 


/dx    ~\  I        ^       X     r    dx  I  .X 


a  a 


*  Forms  7-12  are  preferable  to  the  respective  logarithmic  expressions 
(Art.  19).  on  account  of  the  close  analogy  with  the  circular  forms,  and  also 
because  they  involve  functions  that  are  directly  tabulated.  This  advantage 
appears  more  clearly  in  13-20. 


136  HYPERBOLIC    FUNCTIONS.  [ChAP.  IV. 


).    /  -T-A  =  icoth-^  -,    / 


—  dx  I  ,x 


10.    /  -1 i  =  -coth-^-,    /  -r-7— 1         =  -  cot- 


a^  -\-  x"^  a  a 

II.    /  —  „  =  -sech-'-    / 


dx  I        ,    ^^     r        dx  I  X 

,  =  -sech-  -,    /  — /-,  =  -  sec-'  -,      . 

C      —dx  I        ^       X      r      -  dx  IX 

'^-  J  WFT^'  ==^"'^"  -^^JV7r=7-~a^'^-  -a- 

From  these   fundamental  integrals  the  following  may  be 
derived : 

r        dx  I    .  1,  I  ^•^  +  ^        V         M 

i^.    /  =  — -m  sinh~  — ,  <2:  positive,  <^^><7; 

^    t/    ^ax-j^2bx^c       Va  Vac-b'      ^ 

I         ^     ^  ax  -{-  b  .  .  ,, 

=  — =cosh-  -,  ^positive,  ac<Cb  ', 

Va  Vb'—ac 

I  ^   ax  -\-  b 

=     , cos-  ,  a  negative. 

/dx  I  ax  -\-  b 

—  I         1    1  ^-^  H~  ^  - 

=   ,/T^ tanh      -7==,  ac  <b\  ax  +  b  <  \/b'  —  ac; 

Vb — ac  Vo  —ac 

—  I  ^    ^    ax  -\-  b  „  ,    ,  

=  -7==  coth-^     .^— ,  ac<b\^ax  +  b>  s/b'  -  ac  ; 

Vb  —ac  Vb  —ac 

/»5        dx  1^ 

Thus,    /  -^ r-=-coth~X;^-2)     =coth-'2-coth-'3 

^tanh-X.5)-tanh-\.3333)=. 5494-. 3466=. 2028.^ 

;  .'34^^+3  =-tanh-X^-2)J  =tanh-o-tanh-(.5) 

=  -  .S494. 
(By  interpreting  these  two  integrals  as  areas,  show  graph- 
ically that  the  first  is  positive,  and  the  second  negative.) 


/dx                     2             ,     ,       Ix—b 
/ =     ,  tanh-  A  / r, 
{a-x)Vx-b       Va-b              Aja-b' 


{a—x)  Vx—b 

*For  tanh-i(.5)  interpolate  between  tanh  (.54)  =  .4930,  tanh  (.56)  =  .5080 
fsee  tables,  pp.  162,  163);  and  similarly  for  tanh-i  (.3333). 


Art.  26.J  elementary  integrals.  137 

—  2                ,       I  x—b                 2              ,     ,       Ix'-b 
or      — ,  tan  ~   a  /  -y ,  or      ,_ _   coth-  a  / r  ; 

the  real  form  to  be  taken.     (Put  x  —  b  —  z",  and  apply  9,  10.) 

^      n  dx  2  ,    ,       Ib—x 

ID.     /  —     ,  tanh~  A  /  7 , 

e/    (^— ;i:)  V^— ;tr        4//^— «  V   ^~^ 

,     .       /  b—x  —  2  ,       Ib—x 

-  coth-'  A  /  -7 ,   or       ^  tan      \     ; ; 

a  Y    b—a'  Va-b  y  a-b' 


2 
or 


Vb 
the  real  form  to  be  taken. 

17.  J\x'  -  a'fdx  =  ^x{x'  -  a')^  -  -a'  cosh"'^. 

By  means  of  a  reduction-formula  this  integral  is  easily  made 
to  depend  on  8.  It  may  also  be  obtained  by  transforming 
the  expression  into  hyperbolic  functions  by  the  assumption 
X  =  a  cosh  u,  when  the  integral  takes  the  form 

d'  I  sinh'  udu=  —  I  (cosh  2u  —  i)du  =  -V(sinh  2u  —  2u) 

=  id'{su\h.  u  cosh  u  —  u)y 

which  gives  17  on  replacing  a  cosh  u  by  x,  and  a  sinh  u  by 
{x"^  —' d^)^.  The  geometrical  interpretation  of  the  result  is 
evident,  as  it  expresses  that  the  area  of  a  rectangular-hyper- 
bolic segment  AMP  is  the  difference  between  a  triangle  OMP 
and  a  sector  OAP. 

18.  /{a'  -  x'idx  =  -^x{a^  -  x'"^  +  -a'  sin"'  ^. 
19-  f{x"  +  ci'Ydx  =  -x{x'  +  ^'•)^  +  -a'  sinh-^  -. 

20.  Aec'  0^0  '=y(^  +  ^^"'  ^)^^  ^^"  <5^ 

=  i  tan  0(1  +  tan'  0)^  +  i  sinh"'  (tan  0) 
=  i  sec  0  tan  0  +  -J  gd"^  0. 

21.  /  sech^udu=  -J  sech  z^  tanh  ^^  +  i  gd  21, 

Prob.  71.   What  is  the  geometrical  interpretation  of  18,  19? 
Prob.  72.  Show  that  J  {ax^  -f  2/^jt:  +  <^)^^^  reduces  to  17,  18,  19, 


138  H\P£RBOLIC    FUNCTIONS.  [Chap.  IV. 

respectively:  when  a  is  positive,  with  ac  <  b""  ]  when  a  is  negative  ; 
and  when  a  is  positive,  with  ae>  b"^. 

Prob.  73.  Prove    /  sinh  u  tanh  u  du  —  sinh  z/  —  gd  z^, 

fcosh  u  coth  u  du  =  cosh  u  -f  log  tanh  -. 
Prob.  74.  Integrate 

{x'  +  2Jt:  4-  5) "  Vjc:,       {x'  +  2^  +  5)-V^,      (Jt:'  +  2^  +  5)^^:^. 

Prob.  75.  In  the  parabola  /  =  Ap^,  if  s  be  the  length  of  arc 
measured  from  the  vertex,  and  0  the  angle  which  the  tangent  line 
makes  with  the  vertical  tangent,  prove  that  the  intrinsic  equation  of 
the  curve  is  ds/dcp  =  2p  sec^  <p,  s  —  ^  ^qc  <p  tan  0  +  i  gd~^0- 

Prob.  76.  The  polar  equation  of  a  parabola  being  r  =  a  sec'  ^6,. 
referred  to  its  focus  as  pole,  express  s  in  terms  of  0. 

Prob.  77.  Find  the  intrinsic  equation  of  the  curve jv/«  =  cosh  x/a, 
and  of  the  curve  y/a  =  log  sec  x/a. 

Prob.  78.  Investigate  a  formula  of  reduction  iorj  cosh**^^^;, 

also  integrate  by  parts  cosh~^Jt:  ^^,  t3.nh~'^x  dx,  (sinh'^xydx;    and 

show  that  the  ordinary  methods  of  reduction  for    /  cos'"^sin":x:^x 

can  be  applied  to  /  cosh'''  x  sinh"  x  dx. 

Art.  27.    Functions  of  Complex  Numbers. 

As  vector  quantities  are  of  frequent  occurence  in  Mathe- 
matical Physics ;  and  as  the  numerical  measure  of  a  vector 
in  terms  of  a  standard  vector  is  a  complex  number  of  the 
form  x-\-iy,  in  which  x,  y  are  real,  and  i  stands  for  V—  i;  it 
becomes  necessary  in  treating  of  any  class  of  functional  oper- 
ations to  consider  the  meaning  of  these  operations  when  per- 
formed on  such  generalized  numbers.*  The  geometrical  defini- 
tions of  cosh  7^,  sinh  ?/,  given  in  Art.  7,  being  then  no  longer 
applicable,  it  is  necessary  to    assign  to  each  of  the  symbols 

*The  use  of  vectors  in  electrical  theory  is  shown  in  Bedell  and  Crehore's 
Alternating  Currents,  Chaps,  xiv-xx  (first  published  in  1892).  The  advantage 
of  introducing  the  complex  measures  of  such  vectors  into  the  differential  equa- 
tions is  shown  by  Steinmetz,  Proc.  Elec.  Congress,  1893;  while  the  additional 
convenience  of  expressing  the  solution  in  hyperbolic  functions  of  these  complex 
numbers  is  exemplified  by  Kennelly,  Proc.  American  Institute  Electrical 
Engineers,  April  1895.      (See  below,  Art.  37.) 


Art.  27.]  functions  of  complex  numbers.  139 

cosh  (x  -\-  iy),  sinh  {x  -\-  iy),  a  suitable  algebraic  meaning, 
which  should  be  consistent  with  the  known  algebraic  values  of 
cosh  X,  sinh  ;r,  and  include  these  values  as  a  particular  case 
when  ^  =  o.  The  meanings  assigned  should  also,  if  possible, 
be  such  as  to  permit  the  addition-formulas  of  Art.  1 1  to  be 
made  general,  with  all  the  consequences  that  flow  from  them. 

Such  definitions  are  furnished  by  the  algebraic  develop- 
ments in  Art.  i6,  which  are  convergent  for  all  values  of  u,  real 
or  complex.  Thus  the  definitions  of  cosh  {x  +  iy\  sinh  {x  -\-  iy) 
are  to  be 


cosh  (;«;  +  e»  =  I  +  ^ix  -f  iyy  -f  ^{x  +  iy)'  + 
sinh  [x  +  iy)  =  {x  +  iy)  +  ^(^  +  2»«  -f  . . . 


(52) 


From  these  series  the  numerical  values  of  cosh  {x  -f-  iy), 
sinh  {x -\- iy)  could  be  computed  to  any  degree  of  approxima^ 
tion,  when  x  and  y  are  given.  In  general  the  results  will  come 
out  in  the  complex  form* 

cosh  {x  -\- iy)  =  a -\-  ib, 
sinh  (x  -\-  iy)  =  c  -\-  id. 
The  other  functions  are  defined  as  in  Art.  /,  eq.  (9). 

Prob.  79.  Prove  from  these  definitions  that,  whatever  u  may  be, 
cosh  (—  «)  =  cosh  «,  sinh  (—«)  =  —  sinh  «, 

-r  cosh  u     ■=.  sinh  u.  -r-  sinh  u    =  cosh  u, 

du  ^  du  ' 

-^-icosh  mu  =  m*  cosh  mu,   -—^  sinh  mu  =  m^  sinh  mu.\ 
du^  du"  ' 

*It  is  to  be  borne  in  mind  that  the  symbols  cosh,  sinh,  here  stand  for  alge- 
braic operators  which  convert  one  number  into  another;  or  which,  in  the  lan- 
guage of  vector-analysis  change  one  vector  into  another,  by  stretching  and 
turning. 

\  The  generalized  hyperbolic  functions  usually  present  themselves  in  Mathe- 
matical Physics  as  the  solution  of  the  differential  equation  d'^cp/dti^  =  in^<j>, 
where  0,  tn,  u  are  complex  numbers,  the  measures  of  vector  quantities.  (See 
Art.  37.) 


140  HYPERBOLIC    FUNCTIONS.  [ChAP.  IV. 

Art.  28.    Addition-Theorems  for  Complexes. 

The  addition-theorems  for  cosh  {u  -\-  v),  etc.,  where  u^  v  are 
complex  numbers,  may  be  derived  as  follows.  First  take  u^  v 
as  real  numbers,  then,  by  Art.  ii, 

cosh  {u  -\-v)  —  cosh  u  cosh  v  -\-  sinh  u  sinh  v\ 
hence    i  +  i-,(«  +  vj  +. . .  =(i  +  i^«'  +  . .  .)(i  +  i^z,'+. . .) 

This  equation  is  true  when  u,  v  are  any  real  numbers.  It 
must,  then,  be  an  algebraic  identity.  For,  compare  the  terms 
of  the  rth  degree  in  the  letters  u^  v  on  each  side.     Those  on 

the  left  are  — ;(^+  'vj',  and  those  on  the  right,  when  collected, 

form  an  rth-degree  function  which  is  numerically  equal  to  the 
former  for  more  than  r  values  of  u  when  v  is  constant,  and  for 
more  than  r  values  of  v  when  u  is  constant.  Hence  the  terms 
of  the  rth  degree  on  each  side  are  algebraically  identical  func- 
tions of  u  and  v!^  Similarly  for  the  terms  of  any  other  degree. 
Thus  the  equation  above  written  is  an  algebraic  identity,  and 
is  true  for  all  values  of  u^  v,  whether  real  or  complex.  Then 
writing  for  each  side  its  symbol,  it  follows  that 

cosh  {u  +  v)  =  cosh  u  cosh  v  -\-  sinh  u  sinh  v;  (53) 
and  by  changing  v  into  —  v, 

cosh  {u  —  2>)  =  cosh  u  cosh  v  —  sinh  u  sinh  v.  (54) 

In  a  similar  manner  is  found 

sinh  {u  ±  v)  =  sinh  u  cosh  v  ±  cosh  u  sinh  v.  (55) 

In  particular,  for  a  complex  argument, 

cosh  {x  ±  iy)  =  cosh  x  cosh  ty  ±  sinh  x  sinh  iy,  ) 

}       (56) 
sinh  {x  ±  iy)  =  sinh  x  cosh  ty  ±  cosh  x  sinh  ty.  ) 

*  "  If  two  rth-degree  functions  of  a  single  variable  be  equal  for  more  than  r 
values  of  the  variable,  then  they  are  equal  for  all  values  of  the  variable,  and  are 
algebraically  identical." 


Art.  29.]  FUNCTIONS    OF    PURE    IxMAGINARIES.  141 

Prob.  79.  Show,  by  a  similar  process  of  generalization,*  that  if 
sin  Uy  cos  u^  exp  u\  ht  defined  by  their  developments  in  powers  of 
Uy  then,  whatever  u  may  be, 

sin  {u  -\~  v)  =^  sin  u  cos  v  +  cos  «  sin  Vy 
cos  {u  -\-  v)  =^  cos  u  cos  V  —  sin  u  sin  z;, 
exp  {u  -\-v)  =  exp  ?<!  exp  v. 
Prob.  80.  Prove  that  the  following  are  identities: 
cosh''  u  —  sinh'^  u  =  i^ 
cosh  u  -j-  sinh  z^  =  exp  u^ 
cosh  /^  —  sinh  u  =  exp  (  —  ^), 
cosh  /^  =  i[exp  u  -\-  exp  (  —  /^)], 
sinh  2^  =  i[exp  u —  exp(—  u)]. 

Art.  29.    Functions  of  Pure  Imaginaries. 

In  the  defining  identities 

cosh  u  =  I  -\ — ru^  -| -u*  +  .  . ., 

2!  /}.• 

sinh  u  =  u  -\ — -?/'  -j — ru^  4-  ,  .  . . 

'    3!       '    5!       ' 

put  for  u  the  pure  imaginary  ty,  then 

cosh  2>  =  I  —  -1-/  +  -i  y  -  .  .  .  =  COS7,  (57) 


l!(^»'+i 


sinh  f>  =  />  +  —{iyy  +  -  (/»*  +  .  . 


and,  by  division,      tanh  iy  =  /  tan  y.  (59) 

*  This  method  of  generalization  is  sometimes  called  the  principle  of  the 
"  permanence  of  equivalence  of  forms."  It  is  not,  however,  strictly  speaking,  a 
"  principle,"  but  a  method;  for,  the  validity  of  the  generalization  has  to  be 
demonstrated,  for  any  particular  form,  by  means  of  the  principle  of  the  alge- 
braic identity  of  polynomials  enunciated  in  the  preceding  foot-note.  (See 
Annals  of  Mathematics,  Vol.  6,  p.  81.) 

f  The  symbol  exp  u  stands  for  "exponential  function  of  «, "  which  is  identi- 
cal with  g»  when  u  is  real. 


142 


HYPERBOLIC    FUNCTIONS, 


[Chap.  IV. 


These  formulas  serve  to  interchange  hyperbolic  and  circular 
functions.  The  hyperbolic  cosine  of  a  pure  imaginary  is  real, 
and  the  hyperbolic  sine  and  tangent  are  pure  imaginaries. 

The  following  table  exhibits  tiie  variation  of  sinh  u^  cosh  u, 
tanh  u,  exp  u,  as  u  takes  a  succession  of  pure  imaginary  values. 


u 

sinh  u 

cosh  u 

tanh  u 

exp  u 

O 

O 

I 

0 

I 

lin 

'7i 

.7* 

i 

•7(1+0 

^in 

i 

0 

00  i 

i 

iirr 

'7i 

-•7 

—  i 

.7(1  -  i) 

in 

o 

—  I 

0 

—  I 

iin 

-.7 

-.7(1  +  0 

%in 

—  i 

0 

00  i 

—  i 

\in 

-.yi 

.7 

—  i 

.^{l-^) 

2in 

o 

I 

0 

I 

*  In  this  table  .7  is  written  for  \  1/2,  =  .707  .... 
Prob.  81.  Prove  the  following  identities  : 

cosjF  =  cosh  iy     =  i[exp  iy  +  exp  (—  ?>')], 

sin^  =  -  sinh  ly  =  —[exp  iy  —  exp  (—  ty)]^ 

cos  y  -\-  t  sin  y  =  cosh  ty  +  sinh  I'y  =  exp  ty, 
cos  jv  —  /  sin  jv  =  cosh  ty  —  sinh  iy  =  exp  ( —  ty)^ 
cos  ty  =  cosh  y,     sin  /y  =  /  sinh  jf. 

Prob.  82.  Equating  the  respective  real  and  imaginary  parts  on 
each  side  of  the  equation  cos  ny  -\-  i  sin  ny  =  (cos  y  -\r  i  sin  jv)", 
express  cos  ny  in  powers  of  cos  j,  sin  y  ;  and  hence  derive  the  cor- 
responding expression  for  cosh  ny. 

Prob.  83.  Show  that,  in  the  identities  (57)  and  (58),  y  may  be 
replaced  by  a  general  complex,  and  hence  that 

sinh  {x  ±  iy)  =  ±  i  sin  (y  =F  ix). 


Art.  30.]        functions  of  x  -\-  iy  in  the  form  X  -\-iY.  143 

cosh  {x  ±  iy)  =  cos  {y  ^^  tx), 
sin  (x  ±  iy)  =  ±  i  sinh  (y  =f  ix)j 
cos  [x  ±  iy)  =  cosh  (y  =F  ix). 

Prob.  84.  From   the   product-series    for  sin^   derive  that    for 
rsinh  X  : 

sinh  ^  =  ^(i  +  ^{l  +  ^,)(i  +  ^-^,)  .  .  .. 

Art.  30.    Functions  of  x  +  ijy  in  the  Form  X-{-iY. 
By  the  addition-formulas, 

cosh  {x  -\-  iy)  =  cosh  x  cosh  iy  +  sinh  ;tr  sinh  iy^ 
sinh  (;ir  -|-  iy)  =  sinh  ;tr  cosh  iy  -\-  cosh  ;ir  sinh  iy, 
but  cosh  ?;/  =  cos  y,     sinh  2;/  =  i  sin  7, 

hence    cosh  {x  +  y^)  =  cosh  x  cos  jj^  +  ^  sinh  ;«:  sin  7, 


sinh  {x  -^  iy)  =  sinh  x  cos  j  +  ^  cosh  ;tr  sin  y. 
Thus  if  cosh  (x -\- iy)  =z  a -\-  id,  sinh  {x -\-  iy)  =  ^  +  /<^,  then 
^  =  cosh  X  cos  y,     ^  =  sinh  x  sin  j, 


(61) 
^  =  sinh  X  cos  ^,     d  =  cosh  ;ir  sin  j^. 

From  these  expressions  the  complex  tables  at  the  end  of 
this  chapter  have  been  computed. 

Writing  cosh  2  =Z,  where  2  =  x  -{-  iy,  Z  =  X-\-  iV;  let  the 
•complex  numbers  2,  Z  be  represented  on  Argand  diagrams,  in 
the  usual  way,  by  the  points  whose  coordinates  are  {x,  y), 
{X,  Y) ;  and  let  the  point  z  move  parallel  to  the  j/-axis,  on  a 
given  line  x  —  m,  then  the  point  Z  will  describe  an  ellipse 
whose  equation,  obtained  by  eliminating  y  between  the  equa- 
tions X  ■=  cosh  m  cos  jK,  F=  sinh  m  sin  y,  is 

X^_  F'       _ 

(cosh  tri)^    '    (sinh  mf 

and  which,  as  the  parameter  m  varies,  represents  a  series  of 
•confocal  ellipses,  the    distance  between  whose    foci  is   unity. 


144  HYPERBOLIC    FUNCTIONS.  [ChAP.  IV. 

Similarly,  if  the  point  z  move  parallel  to  the  ;r-axis,  on  a  given 
line  y  =  n,  the  point  Z  will  describe  an  hyperbola  whose  equa- 
tion, obtained  by  eliminating  the  variable  x  from  the  equations 
X=^  cosh  X  cos  n,  Y  =^  sinh  x  sin  n,  is 

X' Y'      _ 

(cos  iif       (sin  nf 

and  which,  as  the  parameter  n  varies,  represents  a  series  of 
hyperbolas  confocal  with  the  former  series  of  ellipses. 

These  two  systems  of  curves,  when  accurately  drawn  at 
close  intervals  on  the  Z  plane,  constitute  a  chart  of  the  hyper- 
bolic cosine ;  and  the  numerical  value  of  cosh  {m  -j-  in)  can  be 
read  off  at  the  intersection  of  the  ellipse  whose  parameter  is  m 
with  the  hyperbola  whose  parameter  is  ?/.* 

Prob.  85.  Prove  that,  in  the  case  of  sinh  {x  -j-  iy)^  the  above  two 

systems  of  curves  are  each  turned  through  a  right  angle.     Compare 

the  chart  of  sin  {x  +  iy),  and  also  of  cos  {x  +  iy). 

sinh  2X  ~4~  /'  sin  2V 

Prob.  86.  Prove  the  identity  tan  ix  +  iy)  = r ; . 

■'         \      '    -^^        ^Q^Yi  2x  +  cos  2y 

Prob.  87.  If  cosh  {x  -\-  iy),  =  a  -\-  ib^  be  written  in  the  "  modulus 
and  amplitude"  form  as  r(cos  d  -\-  i  sin  ^),  —  r  exp  /6',  then 

r"  —  a^  ^  h"  —  cosh''  x  —  sin'jj'  =.cosV  —  sinh'  x^ 
tan  (^  =  b/a  —  tanh  x  tan  y. 

Prob.  88.  Find  the  modulus  and  amplitude  of  sinh  (x  +  iy)^ 
sin  (x  4-  iy)y  exp  {x  +  iy). 

Prob.  89.  The  functions  sinh  u,  cosh  u  have  the  pure  imaginary 
period  2i7r;  that  is,  sinh  (u  +  zin:)  =  sinh  u,  cosh  (u  +  zirr)  =  cosh  z/; 
also  sinh  {u-\-^i7z)  =i  cosh  u,  cosh  {u-{-\in)  =  i  sinh  u,  sinh  {u-\-in) 
=  —  sinh  u,  cosh  {u  -\-  in)  =  —  cosh  u. 

Prob.  90.  The  functions  cosh~Vz,  sinh~V/  have  multiple  values 
at  intervals  of  2/7r,  but  each  has  a  unique  value  (called  the  principal 
value)  in  which  the  coefficient  of  /  lies  between  o  and  tt  for  the 
former,  and  between  —  ^tt  and  +  ^tt  for  the  latter. 

*  Such  a  chart  is  given  by  Kennelly,  Proc.  A.  I.  E.  E.,  April  1895,  and  is 
used  by  him  to  obtain  the  numerical  values  of  cosh  {x-{-tj/),  sinh  (x-{-iy),  which 
present  themselves  as  the  measures  of  certain  vector  quantities  in  the  theory  of 
alternating  currents.  (See  Art.  37.)  The  chart  is  constructed  for  values  of  x 
and  of  jj/  between  o  and  1.2;  but  it  is  available  for  all  values  oly,  on  account  of 
the  periodicity  of  the  functions. 


Art.  31.]  the  catenary.  145 

Art.  31.    The  Catenary. 

A  flexible  inextensible  string  is  suspended  from  two  fixed 
points,  and  takes  up  a  position  of  equilibrium  under  the 
action  of  gravity.  It  is  required  to  find  the  equation  of  the 
curve  in  which  it  hangs. 

Let  w  be  the  weight  of  unit  length,  and  s  the  length  of  arc 
^/*  measured  from  the  lowest  point  A  ;  then  ws  is  the  weight 
of  the  portion  AP.  This  is  balanced  by  the  terminal  tensions, 
T  acting  in  the  tangent  line  at  P,  and  H  in  the  horizontal 
tangent.     Resolving  horizontally  and  vertically  gives 

T'cos  4*^=11^     T  sin  cf)  :=  ws, 
in  which  0  is  the  inclination  of  the  tangent  at  P;  hence 

ws      s 

wheie  c  is  written  for  H/w,  the  length  whose  weight  is  the 
constant  horizontal  tension  ;  therefore 


{fy  _s        ^^  _     /     ^_^^       dx  __       ds 
d^~~c'     ^~"V^  +  ?"'    T"~7F+?' 

—  =  smh-'  — ,  smh  —  =  —  =  y-,  —  =  cosh  — , 
c  c  c        c       ax      c  c 

which  is  the  required  equation  of  the  catenary,  referred  to  an 
axis  of  X  drawn  at  a  distance  c  below  A. 

The  following  trigonometric  method  illustrates  the  use  of 
the  gudermanian  :  The  "  intrinsic  equation,"  s  z=z  c  tan  0, 
gives  ds  =^  c  sec'  0  d(t> ;  hence  dx,  =  ds  cos  (f)f  =  c  sec  0  d(p ; 
^,  =  ^.y  sin  0,=^sec  0  tan  (pdcp;  thus  x=c  gd-*  <p,  y  =  c  sec  0; 
whence  y/c  =  sec  0  =  sec  gd  x/c  =  cosh  x/c ;  and 
s/c  =  tan  gd  x/c  =  sinh  x/c. 

Numerical  Exercise. — A  chain  whose  length  is  30  feet  is 
suspended  from  two  points  20  feet  apart  in  the  same  hori- 
zontal ;  find  the  parameter  c,  and  the  depth  of  the  lowest 
point. 


146  HYPERBOLIC    FUNCTIONS.  [ChAP.  IV. 

The  equation  s/c  =  sinh  x/c  gives  \^/c  =  sinh  lo/c,  which, 
by  putting  lo/c  =  2,  may  be  written  1.5^  =  sinh^.  By  exam- 
ining the  intersection  of  the  graphs  oi  y  =  sinh^,  j/  =  i.^js^ 
it  appears  that  the  root  of  this  equation  is  ^  =  1.6,  nearly. 
To  find  a  closer  approximation  to  the  root,  write  the  equation 
in  the  form/(^)  =  sinh  2  —  i.^z  =  o,  then,  by  the  tables, 

/(1.60)  =  2.2,7  S^  —  2.4000  =  —  .0244, 
/(1.62)  =  2.4276  —  2.4300  =  —  .0024, 
/(1.64)  =  2.4806  —  2.4600  =  +  .0206; 

whence,  by  interpolation,  it  is  found  that  /(1.6221)  =  o,  and 
2  =  1. 622 1,  c  =  \o/z  =.  6.1649.  The  ordinate  of  either  of 
the  fixed  points  is  given   by  the  equation 

yjc  —  cosh  x/c  =  cosh  lo/c  =  cosh  1.6221  =  2.6306, 

from  tables;  hence  j/ =  16.2174,  and  required  depth  of  the 
vertex  =  j/  —  c  =  10.0525  feet.* 

Prob.  91.  In  the  above  numerical  problem,  find  the  inclination 
of  the  terminal  tangent  to  the  horizon. 

Prob.  92.  If  a  perpendicular  MJV  be  drawn  from  the  foot  of  the 
ordinate  to  the  tangent  at  P,  prove  that  MJV  is  equal  to  the  con- 
stant <:,  and  that  JVP  is  equal  to  the  arc  AP.  Hence  show  that 
the  locus  of  iV  is  the  involute  of  the  catenary,  and  has  the  prop- 
erty that  the  length  of  the  tangent,  from  the  point  of  contact  to  the 
axis  of  Xf  is  constant.  (This  is  the  characteristic  property  of  the 
tractory). 

Prob.  93.  The  tension  Tat  any  point  is  equal  to  the  weight  of  a 
portion  of  the  string  whose  length  is  equal  to  the  ordinate  y  of  that 
point. 

Prob.  94.  An  arch  in  the  form  of  an  inverted  catenary  f  is  30 
feet  wide  and  10  feet  high;  show  that  the  length  of  the  arch  can  be 

obtained  from  the  equations  cosh  z  —  —z  =1,       2s  =  -^  sinh  z. 

3  ^ 

*  See  a  similar  problem  in  Chap.  I,  Art.  7. 

f  For  the  theory  of  this  form  of  arch,  see   **  Arch "  in  the  Encyclopaedia 
Britannica. 


Art.  32.]  CATENARY    OF    UNIFORM    STRENGTH.  147 

Art.  32.  Catenary  of  Uniform  Strength. 

If  the  area  of  the  normal  section  at  any  point  be  made 
proportional  to  the  tension  at  that  point,  there  will  then  be  a 
constant  tension  per  unit  of  area,  and  the  tendency  to  break 
will  be  the  same  at  all  points.  To  find  the  equation  of  the 
curve  of  equilibrium  under  gravity,  consider  the  equilibrium  of 
an  element  PP'  whose  length  is  dSy  and  whose  weight  is  gpoodsy 
where  oo  is  the  section  at  P,  and  p  the  uniform  density.  This 
weight  is  balanced  by  the  difference  of  the  vertical  components 
of  the  tensions  at  P  and  P\  hence 

d{  T  sin  0)  =  gfxsjdsy 

d{ T  cos (p)  =  o; 

therefore  T  cos  (p  =  Hy  the  tension  at  the  lowest  point,  and 
T  =  H  sec  0.  Again,  if  gd^  be  the  section  at  the  lowest  point, 
then  by  hypothesis  oo/go^  =  T/H  =  sec  <p,  and  the  first  equation 
becomes 

IId{sec  0  sin  0)  =  gpoo^  sec  (pdSy 

or  c  d  tan  0  =  sec  0  ds, 

where  c  stands  for  the  constant  H/gftoo^,  the  length  of  string 
(of  section  qd^  whose  weight  is  equal  to  the  tension  at  the 
lowest  point ;  hence, 

ds  =  c  sec  0^0,     s/c  —  gd-'0) 
the  intrinsic  equation  of  the  catenary  of  uniform  strength. 

Also     dx  =  ds  cos  0  =  ^  d(p,  dj/  =  ds  sin  0  =  ^  tan  0  d<p ; 

hence  x  =  c<p,  j^  =  c  log  sec  0, 

and  thus  the  Cartesian  equation  is 

j//c  =  log  sec  x/c, 

in  which  the  axis  of  x  is  the  tangent  at  the  lowest  point. 

Prob.  95.  Using  the  same  data  as  in  Art.  31,  find  the  parameter 
r  and  the  depth  of  the  lowest  point.  (The  equation  x/c  =  gd  s/c 
gives     10/ c  =  gd  15/^,    which,    by     putting     15/^  =  z,    becomes 


148  HYPERBOLIC    FUNCTIONS.  [Chap.  IV. 

gd  z  =  ^z.  From  the  graph  it  is  seen  that  z  is  nearly  1.7.  If 
/(^z)  =  gdz  —  %z,  then,  from  the  tables  of  the  gudermanian  at  the 
end  of  this  chapter, 

/(1.70)  =  1. 1780  -  1.1333  =  +  .0447,  Pv 

/(1.75)  =  1. 1796  —  1. 1667  =  +  .0129, 

/(1.80)  =:  1. 1804  —  1.2000  =    —  .0196, 

whence,  by  interpolation,  z  =  1.7698  and  c  =  8.4755.  Again, 
y/^  =■  log«  sec  x/c  ;  but  x/c  =  lo/c  —  1.1799  ;  and  1.1799  radians 
=  67°  36'  29";  hence  y  =  8.4755  X  .41914  X  2.3026  =  8.1798,  the 
required  depth.) 

Prob.  96.  Find  the  inclination  of  the  terminal  tangent. 

Prob.  97.  Show  that  the  curve  has  two  vertical  asymptotes. 

Prob.  98.  Prove  that  the  law  of  the  tension  Z",  and  of  the  section 
Cb3,  at  a  distance  ^,  measured  from  the  lowest  point  along  the 
curve,  is 

T        GO  ,     s 

-=z.=  —  =  cosh  -; 
H      00,  c' 

and  show  that  in  the  above  numerical  example  the  terminal  section 
is  2.85  times  the  minimum  section. 

Prob.  99.  Prove  that  the  radius  of  curvature  is  given  by 
p  ^  c  cosh  s/c.  Also  that  the  weight  of  the  arc  s  is  given  by 
IV=  JI  sinh  s/Cy  in  which  s  is  measured  from  the  vertex. 

Art.  33.    The  Elastic  Catenary. 

An  elastic  string  of  uniform  section  and  density  in  its  natu- 
ral state  is  suspended  from  two  points.  Find  its  equation  of 
equilibrium. 

Let  the  element  do-  stretch  into  ds;  then,  by  Hooke's  law, 
ds  =  d(j{i  -\-  XT),  where  X  is  the  elastic  constant  of  the  string ; 
hence  the  weight  of  the  stretched  element  ds,  =  gpoodcr,  = 
gpoods/{i  +  XT),     Accordingly,  as  before, 

d{T  sm  0)  =z  g poods /{i  -\-  A  J), 

and  T  cos  (f)  =  H  =  gpooc, 

hence  ^^(tan  0)  =  ds/{\  +  /i  sec  0), 

in  which  p.  stands  for  XH,  the  extension  at  the  lowest  point ; 


Art.  34.]  THE  tractory.  149 

therefore        ds  =  <:(sec'  0  +  yu  sec^  (p)d(pf 

s/c  =  tan  0  +  J/<(sec  0  tan  0  +  gd~'  0), 

which  is  the  intrinsic  equation  of  the  curve,  and  reduces  to  that 
of  the  common  catenary  when  yw  —  o.  The  coordinates  ;r,  y 
may  be  expressed  in  terms  of  the  single  parameter  0  by  put- 
ting dx  =  ds  cos  0  =  ^(sec  0  +  /f  sec''  (p)d(p, 

dy  =  ds  sin  (f)  =  <;(sec^  0  +  y"  sec^  0)  sin  0  ^0.     Whence 

x/c  =  gd"'  0  +  >"  tan  0,    j//<;  =  sec  0  +  i/<  tan'  0. 

These  equations   are  more  convenient  than  the  result  of 
eliminating  0,  which  is  somewhat  complicated. 

Art.  34.    The  Tractory.* 

To  find  the  equation  of  the  curve  which  possesses  the 
property  that  the  length  of  the  tangent  from  the  point  of  con- 
tact to  the  axis  of  x  is  con- 
stant. 

Let  Pr,  P'T'  be  two  con- 
secutive tangents  such  that 
PT=P'T'  =  c,  and  let  OT 
=  t\  draw  TS  perpendicular 
XoP'T'\  then  \i  PP' =  ds,  it 
is  evident  that  ST'  differs 
from  ds  by  an  infinitesimal  of  a  higher  order.  Let  P7' make 
an  angle  0  with  OA,  the  axis  of  j/;  then  (to  the  first  order  of 
infinitesimals)  PTdcp  =  TS  =  TT'  cos  0 ;  that  is, 

ccfcp  =  cos  (pdt,     t  =  c  gd~'0> 

X,  =  t  —  c  sin  0,  =  ^(gd~'  0  —  sin  0),    y  =  c  cos  0. 

This  is  a  convenient  single-parameter  form,  which  gives  all 

*  This  curve  is  used  in  Schiele's  anti-friction  pivot  (Minchin's  Statics,  Vol.  i, 
p.  242)  ;  and  in  the  theory  of  the  skew  circular  arch,  the  horizontal  projection 
of  the  joints  being  a  tractory.  (See  "Arch,"  Encyclopaedia  Britannica.)  The 
equation  0  =  gd  t/c  furnishes  a  convenient  method  of  plotting  the  curve. 


150  HYPERBOLIC    FUNCTIONS.  [Chap.  IV. 

values  of  x,  y  2iS  cp  increases  from  o  to  ^tt.     The  value  of  s,  ex- 
pressed in  the  same  form,  is  found  from  the  relation 

ds  =  ST'  =  (it  sin  0  =  <:  tan  0  ^0,     s  ^=^  c  log^  sec  0. 

At  the  point  A,  0  =  o,  x=^o,  5  =  0,   /  =  o,  y^c.      The 
Cartesian  equation,  obtained  by  eliminating  0,  is 


^  =  gd-'  (cos-^  ^)  -  sin  (cos-'  ^J  =  cosh-' J  -  ^i  -■^-. 

If  /^  be  put  for  t/c,  and  be  taken  as  independent  variable, 
0  =  gd  2/,    ;r/<;  =z  u  —  tanh  z/,    _;//^  =  sech  u,    s/c  =  log  cosh  u. 

Prob.  100.  Given  t  =  2c,  show  that  0=  75°  35',  s=  1.3249^, 
^  =  .2658^,  X  =  1.0360^.     At  what  point  is  /  =  ^? 

Prob.  loi.  Show  that  the  evolute  of  the  tractory  is  the  catenary. 
(See  Prob.  92.) 

Prob.  102.  Find  the  radius  of  curvature  of  the  tractory  in  terms 
of  0 ;  and  derive  the  intrinsic  equation  of  the  involute. 

Art.  35.    The  Loxodrome. 

On  the  surface  of  a  sphere  a  curve  starts  from  the  equator 
in  a  given  direction  and  cuts  all  the  meridians  at  the  same 

angle.  To  find  its  equation 
in  latitude-and-longitude  co- 
ordinates : 

Let  the  loxodrome  cross 
two    consecutive    meridians 
AM,  AN  in  the  points  P,  Q; 
let  PR  be  a  parallel  of  lati- 
^^^  tude  ;  let  0M=  x,  MP  =  y, 

MN  =^  dx,  RQ  =  dy,  all  in  radian  measure  ;  and  let  the  angle 
MOP=RPQ  =  a',   then 

tan  a  =  RQ/PR,    but     PR  =  MN  cos  MP* 

hence  dx  tan  a  =^  dy  sec  j^,  and  x  tan  a  =  gd"'  y,  there  being 
no  integration-constant  since  ^  vanishes  with  x;  thus  the  re- 
quired equation  is 

^  =  gd  (jr  tan  a). 

*  Jones,  Trigonometry  (Ithaca,  1890),  p.  185. 


Art.  36.]  combined  flexure  and  tension.  151 

To  find  the  length  of  the  arc  0P\  Integrate  the  equation 

ds  =^  dy  CSC  or,     whence  s  =^  y  esc  a. 

To  illustrate  numerically,  suppose  a  ship  sails  northeast, 
from  a  point  on  the  equator,  until  her  difference  of  longitude  is 
45°,  find  her  latitude  and  distance  : 

Here  tan  a  =  i,  andj^  =  gd  ;i:  =  gd  i;r  =  gd(.7854)  =  .7152 
radians:  s  =iyV2  =■  1.0114  radii.  The  latitude  in  degrees  is 
40.980. 

If  the  ship  set  out  from  latitude  y^,  the  formula  must  be 
modified  as  follows :  Integrating  the  above  differential  equa- 
tion between  the  limits  (^,,  Ji)  and  (x^,  y^  gives 

(^, -^i)tan  «  =  gd->,  -  gd->,; 

hence  gd"  'j,  =  gd  ~'j,  -|-  {x^  —  x^  tan  ^,  from  which  the  final 
latitude  can  be  found  when  the  initial  latitude  and  the  differ- 
ence of  longitude  are  given.  The  distance  sailed  is  equal  to 
(/a  ~ y^  CSC  a  radii,  a  radius  being  60  X  i8o/;r  nautical  miles. 
Mercator's  Chart. — In  this  projection  the  meridians  are 
parallel  straight  lines,  and  the  loxodrome  becomes  the  straight 
line  y  =  X  tan  a,  hence  the  relations  between  the  coordinates  of 
corresponding  points  on  the  plane  and  sphere  are  x'  =  x, 
y'  =  gd~y.  Thus  the  latitude  y  is  magnified  into  gd  ~  jj/,  which 
is  tabulated  under  the  name  of  "  meridional  part  for  latitude 
y** ;  the  values  of  j^  and  o(  y^  being  given  in  minutes.  A  chart 
constructed  accurately  from  the  tables  can  be  used  to  furnish 
graphical  solutions  of  problems  like  the  one  proposed  above. 

Prob.  103.  Find  the  distance  on  a  rhumb  line  between  the  points 
(30°  N,  20°  E)  and  (30°  S,  40°  E). 

Art.  36.    Combined  Flexure  and  Tension. 

A  beam  that  is  built-in  at  one  end  carries  a  load  P  at  the 
other,  and  is  also  subjected  to  a  horizontal  tensile  force  Q  ap- 
pHed  at  the  same  point;  to  find  the  equation  of  the  curve 
assumed  by  its  neutral  surface :  Let  x,  y  be  any  point  of  the 


152  HYPERBOLIC    FUNCTIONS.  [Chap.  IV. 

elastic  curve,  referred  to  the  free  end  as  origin,  then  the  bend- 
ing moment  for  this  point  is  Qy  —  Px,  Hence,  with  the  usual 
notation  of  the  theory  of  flexure,*  ' 

which,  on  putting/  —  mx  =  u,  anddy/i/x^  =  d^u/dx^,  becomes 
d'u 

whence  u  =-  A  cosh  nx  -{-  B  sinh  nx^ 

that  is,  y  =  mx  -\-  A  cosh  nx  -\-  B  sinh  nx. 

The  arbitrary  constants  A,  B  are  to  be  determined  by  the 
terminal  conditions.  At  the  free  end  x  =  Oyy=zO'^  hence  ^ 
must  be  zero,  and 

y  =  mx  -\-  B  sinh  nx, 

-^  =z  m-\-  nB  cosh  nx  ; 
dx 

but  at  the  fixed  end,  x  =  I,  and  dy/dx  =  o,  hence 

B  =  —  m/n  cosh  «/, 

and  accordingly 

m  sinh  nx 

y  =  mx r — 7- 

-^  n  cosh  nl 

To  obtain  the  deflection  of  the  loaded  end,  find  the  ordinate 
of  the  fixed  end  by  putting  ;tr  =  /,  giving 

deflection  =  mil tanh  nl\ 

^        n  ' 

Prob.  104.  Compute  the  deflection  of  a  cast-iron  beam,  2X2 
inches  section,  and  6  feet  span,  built-in  at  one  end  and  carrying 
a  load  of  100  pounds  at  the  other  end,  the  beam  being  subjected 
to  a  horizontal  tension  of  8000  pounds.  [In  this  case  /  =  4/3, 
^  =  15  X  10",  Q  =  8000,  B  =  100  ;  hence  n  =  1/50,  m  =  1/80, 
deflection  =  ■5V(72  —  50  tanh  1.44)  =  ^V(72  —  44-69)  =  .34.1  inches.] 

*  Merriman,  Mechanics  of  Materials  (New  York,  1895),  pp.  70-77,  267-269. 


Art.  37.]  ALTERNATING    CURRENTS.  153 

Prob.  105.  If  the  load  be  uniformly  distributed  over  the  beam, 
say  w  per  linear  unit,  prove  that  the  differential  equation  is 

EI^,  =  Qy-h^x\     or    ^1  =  n\y  -  mx% 

2ftt 

and  that  the  solution  is^  =  ^  cosh  nx-\-  B  sinh  nx  +  mx''  -\ — j-. 

n 

Show  also  how  to  determine  the  arbitrary  constants. 


Art.  37.    Alternating  Currents.* 

In  the  general  problem  treated  the  cable  or  wire  is  regarded 
as  having  resistance,  distributed  capacity,  self-induction,  and 
leakage ;  although  some  of  these  may  be  zero  in  special 
cases.  The  line  will  also  be  considered  to  feed  into  a  receiver 
circuit  of  any  description ;  and  the  general  solution  will  in- 
clude the  particular  cases  in  which  the  receiving  end  is  either 
grounded  or  insulated.  The  electromotive  force  may,  without 
loss  of  generality,  be  taken  as  a  simple  harmonic  function  of 
the  time,  because  any  periodic  function  can  be  expressed  in  a 
Fourier  series  of  simple  harmonics. f  The  E.M.F.  and  the 
current,  which  may  differ  in  phase  by  any  angle,  will  be 
supposed  to  have  given  values  at  the  terminals  of  the  receiver 
circuit ;  and  the  problem  then  is  to  determine  the  E.M.F. 
and  current  that  must  be  kept  up  at  the  generator  terminals  ; 
and  also  to  express  the  values  of  these  quantities  at  any  inter- 
mediate point,  distant  x  from  the  receiving  end  ;  the  four 
line-constants  being  supposed  known,  viz.: 

R  =:  resistance,  in  ohms  per  mile, 

L  —  coefficient  of  self-induction,  in  henrys  per  mile, 

C  =  capacity,  in  farads  per  mile, 

G  =  coefficient  of  leakage,  in  mhos  per  mile.J 

It  is  shown  in  standard  works  §  that  if  any  simple  harmonic 

*  See  references  in  foot-note  Art.  27.  f  Chapter  V,  Art.  8. 

X  Kennelly  denotes  these  constants  by  r,  I,  c,  g.  Steinmetz  writes  s  for 
<»Z,  K"  for  aaC,  6  for  G,  and  he  uses  Cfor  current. 

§  Thomson  and  Tait,  Natural  Philosophy,  Vol,  I.  p.  40;  Rayleigh,  Theory 
of  Sound,  Vol.  I.  p.  20;  Bedell  and  Crehore,  Alternating  Currents,  p.  214. 


154  HYPERBOLIC    FUNCTIONS.  [Chap.  IV. 

function  a  sin  {oot  -\-  6)  be  represented  by  a  vector  of  length 
a  and  angle  6,  then  two  simple  harmonics  of  the  same  period 
27r/G0,  but  having  different  values  of  the  phase-angle  8,  can  be 
combined  by  adding  their  representative  vectors.  Now  the 
E.M.F.  and  the  current  at  any  point  of  the  circuit,  distant  x 
from  the  receiving  end,  are  of  the  form 

e  =  e^  sin  {aot  -\-0),     i  =  2,  sin  {cot  +  6\  (64) 

in  which  the  maximum  values  ^j,  i^y  and  the  phase-angles  6,  d\ 
are  all  functions  of  x.  These  simple  harmonics  will  be  repre- 
sented by  the  vectors  ejd,  ijd' ;  whose  numerical  measures 
are  the  complexes  e^  (cos  6  +y  sin  6^)*,  i^  (cos  6'  -{-j  sin  6'), 
which  will  be  denoted  by  /,  t.  The  relations  between  e  and  i 
may  be  obtained  from  the  ordinary  equations  f 

for,  since  de/dt  =  ooe,  cos  {oot  +  ^)  =  ooe^  sin  {oot  -\-  0  -\-  j-7r), 
then  de/dt  will  be  represented  by  the  vector  ooe^  /^  +  i^  ;  and 
di/dx  by  the  sum  of  the  two  vectors  Ge^  /0,  Caoe^  /O  -\-  ^n ; 
whose  numerical  measures  are  the  complexes  Ge,joDCe;  and 
similarly  for  de/dx  in  the  second  equation  ;  thus  the  relations 
between  the  complexes  ^,  i  are 

f^  =  {G+jc.C)e-,      %  =  {R+ja,L)u  m)X 

*In  electrical  theory  the  symbol  7  is  used,  instead  of  i,  for  |/—  i. 

f  Bedell  and  Crehore,  Alternating  Currents,  p.  181.  The  sign  of  dx  is 
changed,  because  x  is  measured  from  the  receiving  end.  The  coefl5cient  of 
leakage,  G,  is  usually  taken  zero,  but  is  here  retained  for  generality  and  sym- 
metry. 

X  These  relations  have  the  advantage  of  not  involving  the  time.  Steinmetz 
derives  them  from  first  principles  vi^ithout  using  the  variable  t.  For  instance, 
he  regards  R  -\-  JooL  as  a  generalized  resistance-coefficient,  vsrhich,  when  applied 
to  i,  gives  an  E.M.F.,  part  of  which  is  in  phase  with  i,  and  part  in  quadrature 
with  i.  Kennelly  calls  R  -\-  JGoL  the  conductor  impedance;  and  G  -\-  jcoC  the 
dielectric  admittance;  the  reciprocal  of  which  is  the  dielectric  impedance. 


Art.  37.]  alternating  currents.  155 

Differentiating  and  substituting  give 


^  =  {RJ^j,^L){G^jmC)e-, 


(67) 


and  thus  /,  i  are  similar  functions  of  Xy  to  be  distinguished  only 
by  their  terminal  values. 

It  is  now  convenient  to  define  two  constants  w,  m^  by  the 
equations* 

m'  =  {R  +jGoL){G  +JG0CI  m:  =  (R  +jgdL)/{G  +J00C) ;  (68) 

and  the  differential  equations  may  then  be  written 

the  solutions  of  which  are  f 

e  =  A  cosh  mx  -{-  B  sinh  mx,     i  =  A^  cosh  mx  -{-  B'  sinh  mx^ 

wherein  only  two  of  the  four  constants  are  arbitrary ;  for  sub- 
stituting in  either  of  the  equations  (66),  and  equating  coeffi- 
cients, give 

{G  ^jodC)  a  =  mB\     (G  +jojC)B  =  mA\ 
whence  B'  =  A/m,,     A'  =  B/m,, 

Next  let  the  assigned  terminal  values  of  e,  i,  at  the  receiver, 
be  denoted  by  E,  /;  then  putting.^  =  o  gives  E  =  A,  /=  A\ 
whence  B  =  mj,  B'  =  E/m^ ;  and  thus  the  general  solution  is 


e  =^  E  cosh  mx  +  mj  sinh  mx^ 

I  _ 
1  =  1  cosh  mx  -f-  ~E  sinh  mx. 


(70) 


*  The  complex  constants  m,  Wi,   are  written  z,  y  by  Kennelly;   and    the' 
variable  length  x  is  written  Za.     Steinmetz  writes  v  for  m. 
f  See  Art.  14,  Probs.  28-30;  and  Art.  27,  foot-note. 


156  HYPERBOLIC    FUNCTIONS.  [Chap.  IV. 

If  desired,  these  expressions  could  be  thrown  into  the  ordi- 
nary complex  form  X-\-jVj  X'  -{-jV\  by  putting  for  the  let- 
ters their  complex  values,  and  applying  the  addition-theorems 
for  the  hyperbolic  sine  and  cosine.  The  quantities  X,  F,  X\ 
Y'  would  then  be  expressed  as  functions  of  x ;  and  the  repre 
sentative  vectors  of  e,  i,  would  be  e^  /6,  i^  /6\  where  e^  z=X''-\-  Y\ 
•.  ^  j^n  j^  Y'\  tan  d  =  V/X,  tan"^  ^It'yX. 

For  purposes  of  numerical  computation,  however,  the  for- 
mulas (70)  are  the  most  convenient,  when  either  a  chart,*  or  a 
table,f  of  cosh  u,  sinh  u,  is  available,  for  complex  values  of  u. 

Prob.  106. J  Given  the  four  line-constants:  J^  =  2  ohms  per  mile, 
Z=  20  millihenrys  per  mile,  C=  1/2  microfarad  per  mile,  G  =  o; 
and  given  go,  the  angular  velocity  of  E.M.F.  to  be  2000  radians 
per  second;  then 

GoZ  =  40  ohms,  conductor  reactance  per  mile; 
i?  -{-jGoL  =  2  +  40J  ohms,  conductor  impedance  per  mile; 

GoC  =  .001  mho,  dielectric  susceptance  per  mile; 
G  -\-  jooC  =  .00 1/'  mho,  dielectric  admittance  per  mile; 
<(^Q-^jGoC)~^  =  —  looq;'  ohms,  dielectric  impedance  per  mile; 

m"^  =  [R  -{-  jGoL){G  -{-JgdC)  —  .04  +  .0027,  which  is  the 
measure  of  .04005  /177"  8^;  therefore 

m  =  measure  of  .2001  /88°  34^  =  .0050  +  .2000/,  an    ab- 
stract coefficient  per  mile,  of  dimensions  [length]  ~S 
mm^  =  m/(G  -{-  j'goC)  =  200  —  5/'  ohms  per  mile. 

Next  let  the  assigned  terminals  conditions  at  the  receiver  be: 
1=0  (line  insulated) ;  and  E  =  1000  volts,  whose  phase  may  be  taken 
as  the  standard  (or  zero)  phase;  then  at  any  distance  x,  by  (70), 

e  ^  E  cosh  mx,      i  =  - —  sinh  mx, 

in  which  mx  is  an  abstract  complex. 

Suppose  it  is  required  to  find  the  E.M.F.  and  current  that  must 
be  kept  up  at  a  generator  100  miles  away;  then 

*  Art.  30,  foot-note.  t  See  Table  II. 

X  The  data  for  this  example  are  taken  from  Kennelly's  article. 


Art.  37.J  alternating  currents.         '  157 

^=  looo  cosh  (.5  -|-  20/),     I  =  200(40  —  j)~^  sinh  (.5  -f  20;'), 
but,  by  Prob.  89,  cosh  (.5  +  20/')  =  cosh  (.5  +  20;*  —  67tj)      ^  ._ 

=  cosh  (.5  +  1. 15;)  =  .4600  +  .4750;' 

obtained  from  Table  II,  by  interpolation  between  cosh  (.5  +  i.iy) 
and  cosh  (.5  +  1.27);  hence 

e  —  460  -|-  475/"  =  ^j(cos  d  -j-/  sin  6)^ 

where   log  tan    d  =  log  475  —  log  460  =  •0139,  0  =  45°    55',     and 
e^  =  460  sec  0=  625.9  volts,  the  required  E.M.F. 

Similarly  sinh  (.5  +  207)  =  sinh  (.5  +  1.157)  =  .2126  +  1.0280;^ 
and  hence 

I  =  — — (100  +y)(.2i26  + 1.0287)  —  ~7 — (4046  +  20607) 

=  /;(cos  6/' +7  sin  <9'), 
where  log  tan  0'  =  9.70684,  6'=  26°  59',  /j  =  4046  sec  ^'/i6oi  =  2.77 
amperes,  the  phase  and  magnitude  of  required  current. 

Next  let  it  be  required  to  find  e  bX  x  =■  2>\  then 

e  =.  1000  cosh  (.04  +  1.67)  =  10007  sinh  (.04  -|-  .037), 

by   subtracting   ^nj\  and   applying   Prob.   89.       Interpolation  be- 
tween sinh  (o  +  07)  and  sinh  (o  +  .17)  gives 

sinh  (o  -[-  .037)  =  00000  -\-  .029957'. 

Similarly  sinh  (.1  -f  -037)  =  .10004  +  •03004/« 

Interpolation  between  the  last  two  gives 

sinh  (.04  -\-  .037)  =  .04002  +  .02999/ 

Hence/ =7(40.02  +29.997)=  —29.99+40.027  =^,(cos  ^+7  sin  6^), 
where 

log  tan  6  =  .12530,  6=  126°  51',^!  =  —  29.99  sec  126°  51'  =  50.01 
volts. 

Again,  let  it  be  required  to  find  ^  at  jjc  =  16;  here 

e  =  1000  cosh  (.08  +  3.27)  =  —  1000  cosh  (.08  +  .067), 

but  cosh  (o  +  .067)  =  .9970  +  07,  cosh  (.1  +  .067)  =  1.0020  +  .0067^ 

hence  cosh  (.08  +  .067)  =  1.0010  +.00487, 

and  e=  —  1001+4.87  =  ^,(cos  6^ +7  sin  0), 

where  6  =  180°  17',  e^  =  looi  volts.     Thus  at  a  distance  of  about 
16  miles  the  E.M.F.  is  the  same  as  at  the  receiver,  but  in  opposite 


158  HYPERBOLIC    FUNCTIONS.  [ChAP.  IV. 

phase.  Since  e  is  proportional  to  cosh  (.005  A-  .2j)x,  the  value  of 
X  for  which  the  phase  is  exactly  180°  is  7r/.2  =  15.7.  Similarly 
the  phase  of  the  E.M.F.  2i\.  x  =^  7.85  is  90°.  There  is  agreement 
in  phase  at  any  two  points  whose  distance  apart  is  31.4  miles. 

In  conclusion  take  the  more  general  terminal  conditions  in 
which  the  line  feeds  into  a  receiver  circuit,  and  suppose  the  current 
is  to  be  kept  at  50  amperes,  in  a  phase  40°  in  advance  of  the  elec- 
trumotive  force;  then  /—  5o(cos  40°  +/  sin  40°)  =  38.30  +  32.14/; 
and  substituting  the  constants  in  (70)  gives 

e—  1000  cosh  (.005  -j-  'y')x  +  (7^21  -|-  62367)  sinh  (.005  -j-  .2j)x 
=  460+4757 -4748+93667= -4288+98417=^^ (cos  0-\-js'm  6)^ 

where  0=  113°  33',  e^  =  10730  volts,  the  E.M.F.  at  sending  end. 
This  is  17  times  what  was  required  when  the  other  end  was  insulated. 
Prob.  107.  If  the  receiving  end  be  grounded,  that  is  if  ^=0; 
and  if  a  current  of  10  amperes  be  caused  to  flow  to  ground;  find 
the  E.M.F.  and  current  to  be  kept  up  at  the  generator.  Also 
compute  these  quantities,  and  their  phases,  at  the  distances  7.85, 
15.7,  31.42,  94.25  miles  from  the  receiver. 

Prob.  108.  If  self-induction  and  capacity  be  zero,  and  the 
receiving  end  be  insulated,  show  that  the  graph  of  the  electromo- 
tive force  is  a  catenary. 

Prob.  T09.  Neglecting  leakage  and  capacity,  prove  that  the 
solution  of  equations  (66)  \s  T  —  I,  e  =  E  -\-  {R  -\- jgoL)Ix. 

Prob.  no.  If  x  be  measured  from  the  sending  end,  show  how 
equations  (65),  (66)  are  to  be  modified;  and  prove  that 

_  _  I    _ 

i  =  E^  cosh  mx  —  mj^  sinh  mx^     ^  =  A  cosh  mx  —  ~E^  sinh  mXj 

where  E^  I^  refer  to  the  sending  end. 

Art.  38.    Miscellaneous  Applications. 

1.  The  length  of  the  arc  of  the  logarithmic  curve  y  =^  a^  is 
s=  ^(cosh  11  +  log  tanh  ^u),  in  vi^hich  M=  i/log  a,  s\nhu=  y/M. 

2.  The  length  of  arc  of  the  spiral  of  Archimedes  r  =  ^6^  is 
s  =  ^^(sinh  2u  +  2u),  where  sinh  u=  0. 

3.  In  the  hyperbola  x^/a'  —  y/<^'  =  i  the  radius  of  curva- 
ture is  p  =  {a''  sinh'  u  +  ^'  cosh'  uf/ab ;  in  which  u  is  the 
measure  of  the  sector  A  OP,  i.e.  cosh  u  =  x/a,  sinh  u  ■=^y/b. 

4.  In  an  oblate  spheroid,  the  superficial  area  of  the  zone 


Art.  88.]  MISCELLANEOUS    APPLICATIONS.  159 

between  the  equator  and  a  parallel  plane  at  a  distance  y  is 
5  —  7r<^'(sinh  2u  +  2u)/2e,  wherein  b  is  the  axial  radius,  e  the 
eccentricity,  u  =  ey/p,  and/  parameter  of  generating  ellipse. 

5.  The  length  of  the  arc  of  the  parabola/'  =  2px,  measured 
from  the  vertex  of  the  curve,  is  /  =  i/(sinh  2u-\-2u),  in  which 
sinh  u  =y/p  =  tan  0,  where  0  is  the  inclination  of  the  termi- 
nal tangent  to  the  initial  one. 

6.  The  centre  of  gravity  of  this  arc  is  given  by 

3/;r  =/'(cosh'  u  —  i\     6^ly  =  p-'  (sinh  411  —  4?/)  ; 

and  the  surface  of  a  paraboloid  of  revolution  is  5=  2  7r yl. 

7.  The  moment  of  inertia  of  the  same  arc  about  its  ter. 
minal  ordinate  is  T  =  pi[xl{x  —  2x)  + -^p^N\  where  //  is 
the  mass  of  unit  length,  and 

N=  u  —  \  sinh  2u  —  \  sinh  /^u  +  -^  sinh  6u. 

8.  The  centre  of  gravity  of  the  arc  of  a  catenary  measured 
from  the  lowest  point  is  given  by 

4/^=  i:'(sinh  2u  -\-  2u)\  Ix  =  c^{u  sinh  u  —  cosh  u  +  l), 

in  which  u  =  x/c ;  and  the  moment  of  inertia  of  this  arc  about 
its  terminal  abscissa  is 

/  =  //^"(tV  s'"^  V^  +  f  sinh  u  —  u  cosh  ti), 

9.  Applications  to  the  vibrations  of  bars  are  given  in  Ray- 
leigh,  Theory  of  Sound,  Vol.  I,  art.  170;  to  the  torsion  of 
prisms  in  Love,  Elasticity,  pp.  166-74;  to  the  flow  of  heat 
and  electricity  in  Byerly,  Fourier  Series,  pp.  75-81  ;  to  wave 
motion  in  fluids  in  Rayleigh,  Vol.  I,  Appendix,  p.  477,  and  in 
Bassett,  Hydrodynamics,  arts.  120,  384;  to  the  theory  of 
potential  in  Byerly  p.  135,  and  in  Maxwell,  Electricity,  arts. 
172-4;  to  Non-Euclidian  geometry  and  many  other  subjects 
in  Giinther,  Hyperbelfunktionen,  Chaps.  V  and  VI.  Several 
numerical  examples  are  worked  out  in  Laisant,,Essai  sur  les 
fonctions  hyperboliques. 


160  HYPERBOLIC    FUNCTIONS.  [Chap.  IV. 

Art.  39.    Explanation  of  Tables. 

In  Table  I  the  numerical  values  of  the  hyperbolic  functions 
sinh  u,  cosh  u,  tanh  u  are  tabulated  for  values  of  it  increasing 
from  o  to  4  at  intervals  of  .02.  When  u  exceeds  4,  Table  IV 
may  be  used. 

Table  II  gives  hyperbolic  functions  of  complex  arguments, 
in  which 

cosh  (^  ±  2»  =  a  ±  ib,     sinh  (x  -^iy^zzic  ±_  id, 

and  the  values  of  ^,  b,  c,  d  are  tabulated  for  values  of  x 
and  oi  y  ranging  separately  from  o  to  1.5  at  intervals  of  .1. 
When  interpolation  is  necessary  it  may  be  performed  in  three 
stages.  For  example,  to  find  cosh  (.82  +  1.342) :  First  find 
cosh  (.82 -[-  i-30»  by  keeping  J  at  1.3  and  interpolating  between 
the  entries  under  ;ir  =  .8  andjc  =  .9  ;  next  find  cosh  (.82  -j-  1-40^ 
by  keeping  jj/  at  1.4  and  interpolating  between  the  entries  under 
;r  =  .8  and  x  =  .9,  as  before ;  then  by  interpolation  between 
cosh  (.82+  1.32)  and  cosh  (.824-  1.42)  find  cosh(  .82  +  1.340. 
in  which  x  is  kept  at  .82.  The  table  is  available  for  all  values 
of  J,  however  great,  by  means  of  the  formulas 

sinh  {x  +  2t7r )  =  sinh;tr,     cosh  (x-\-  2i7i)  =  cosh  x,  etc. 

It  does  not  apply  when  x  is  greater  than  1.5,  but  this  case  sel- 
dom occurs  in  practice.  This  table  can  also  be  used  as  a  com- 
plex table  of  circular  functions,  for 

cos  (y  ±  ix)  =  a^  ib,     sin  {y  ±  ix)  =  d ±u; 

and,  moreover,  the  exponential  function  is  given  by 

exp  {±  X  ±  iy)  =  a  ±  c  ±  i(b  ±  d), 

in  which  the  signs  of  c  and  ^^Tare  to  be  taken  the  same  as  the 
sign  of  X,  and  the  sign  of  z  on  the  right  is  to  be  the  product  of 
the  signs  of  x  and  of  z  on  the  left. 

Table  III  gives  the  values  of  v  =  gd  u,  and  of  the  guder- 
manian  angle  ^=  i8o°  v/n,  as  u  changes  from  o  to  i   at  inter- 


Art.  39.]        /  explanation  of  tables.     '  161 

vals  of  .02/ from  i  to  2  at  intervals  of  .05,  and  from  2  to  4  at 
intervals  of  .1. 

In  Tible  IV  are  given  the  values  of  gd  Uy  log  sinh  u,  log 
cosh  Uy  ^s  u  increases  from  4  to  6  at  intervals  of  .1,  from  6  to 
7  at  intervals  of  .2,  and  from  7  to  9  at  intervals  of  .5. 

In  the  rare  cases  in  which  more  extensive  tables  are  neces- 
sary, reference  may  be  made  to  the  tables^  of  Gudermann, 
Glaisher,  and  Geipel  and  Kilgour.  In  the  first  the  Guderman- 
ian  angle  (written  k)  is  taken  as  the  independent  variable,  and 
increases  from  o  to  100  grades  at  intervals  of  .01,  the  corre- 
sponding value  of  u  (written  Lk)  being  tabulated.  In  the  usual 
case,  in  which  the  table  is  entered  with  the  value  of  Uy  it  gives 
by  interpolation  the  value  of  the  gudermanian  angle,  whose 
circular  functions  would  then  give  the  hyperbolic  functions 
of  71.  When  u  is  large,  this  angle  is  so  nearly  right  that  inter- 
polation is  not  reliable.  To  remedy  this  inconvenience  Gu- 
dermann's  second  table  gives  directly  log  sinh  u,  log  cosh  Uy 
log  tanh  Uy  to  nine  figures,  for  values  oiu  varying  by  .001  from 
2  to  5,  and  by  .01  from  5  to  12. 

Glaisher  has  tabulated  the  values  of  e^  and  ^~*,  to  nine  sig- 
nificant figures,  as  x  varies  by  .001  from  o  to  .1,  by  .01  from  o 
to  2,  by  .1  from  o  to  10,  and  by  I  from  o  to  500.  From  these 
the  values  of  cosh  Xy  sinh  x  are  easily  obtained. 

Geipel  and  Kilgour's  handbook  gives  the  values  of  cosh;r, 
sinh  Xy  to  seven  figures,  as  x  varies  by  .01  from  o  to  4. 

There  are  also  extensive  tables  by  Forti,  Gronau,  Vassal, 
Callet,  and  Hoiiel ;  and  there  are  four-place  tables  in  Byerly's 
Fourier  Series,  and  in  Wheeler's  Trigonometry. 

In  the  following  tables  a  dash  over  a  final  digit  indicates 
that  the  number  has  been  increased. 

*  Gudermann  in  Crelle's  Journal,  vols.  6-9,  1831-2  (published  separately 
under  the  title  Theorie  der  hyperbolischen  Functionen,  Berlin,  1833).  Glaisher 
in  Cambridge  Phil.  Trans.,  vol.  13,  1881.  Geipel  and  Kilgour's  Electrical  Hand- 
book. 


162 


HYPERBOLIC    FUNCTIONS. 


[Chap.  IV. 


Table  I. — Hyperbolic  Functions. 


u. 

sinh  u. 

cosh  «. 

tanh  «. 

u. 

sinh  «. 

cosh  u. 

tanh  u. 

.00 

.0000 

1.0000 

.0000 

1.00 

1.1752 

1.5431 

.7616 

02 

0200 

1.0020 

0200 

1.02 

1.2063 

1.5669 

7699 

04 

0400 

1.0008 

0400 

1.04 

1.2379 

1.5913 

7779 

06 

0600 

1.0018 

0599 

1.06 

1.2700 

1.6164 

7857 

08 

0801 

1.0032 

0798 

1.08 

1.3025 

1.6421 

7932 

.10 

.1002 

1.0050 

.0997 

1.10 

1.3356 

1.6685 

.8005 

12 

1203 

1  0072 

1194 

1.12 

1.3693 

1.6956 

8076 

14 

1405 

1.0098 

1391 

1.14 

1.4035 

1.7233 

8144 

16 

1607 

1.0128 

1586 

1.16 

1.4382 

1.7517 

8210 

18 

1810 

1.0162 

1781 

1.18 

1.4735 

1.7808 

8275 

.20 

.2013 

1.0201 

.1974 

1.20 

1.5095 

1.8107 

.8337 

22 

2218 

1.0243 

2165 

1.22 

1.5460 

1.8412 

8397 

24 

2423 

1.0289 

2355 

1.24 

1.5831 

1.8725 

8455 

26 

2629 

1.0340 

2543 

1.26 

1.6209 

1.9045 

8511 

28 

2837 

1.0395 

2729 

1.28 

1.6593 

1.9373 

8565 

-30 

.3045 

1.0453 

.2913 

1.30 

1.6984 

1.9709 

.8617 

32 

3255 

1.0516 

3095 

1.32 

1.7381 

2.0053 

8668 

M 

3466 

1.0584 

3275 

1.34 

1.7786 

2.0404 

8717 

36 

3678 

1.0655 

3452 

1.36 

1.8198 

2.0764 

8764 

38 

3892 

1.0731 

3627 

1.38 

1.8617 

2.1132 

8810 

.40 

.4108 

1.0811 

.3799 

1.40 

1.9043 

2  1509 

.8854 

42 

4325 

1.0895 

3969 

1.42 

1.9477 

2.1894 

8896 

44 

4543 

1.0984 

4136 

1.44 

1.9919 

2.2288 

8937 

46 

4764 

1.1077 

4301 

1.46 

2.0369 

2.2691 

8977 

48 

4986 

1.1174 

4462 

1.48 

2.0827 

2.3103 

9015 

.50 

.5211 

1.1276 

.4621 

1.50 

2.1293 

2.3524 

.9051 

52 

5438 

1.1383 

4777 

1.52 

2.1768 

2.3955 

9087 

54 

5666 

1.1494 

4930 

1.54 

2.2251 

2.4395 

9121 

56 

5897 

1.1609 

5080 

1.56 

2.2748 

2.4845 

9154 

58 

6131 

1.1730 

5227 

1.58 

2.3245 

2.5305 

9186 

,60 

.6367 

1.1855 

.5370 

1.60 

2.3756 

2.5775 

.9217 

62 

6605 

1.1984 

5511 

1.62 

2.4276 

2  6255 

9246 

64 

6846 

12119 

5649 

1.64 

2.4806 

2.6746 

9275 

66 

7090 

1.2258 

5784 

1.66 

2.5346 

2.7247 

9302 

68 

7336 

1.2402 

5915 

1.68 

2.5896 

2.7760 

9329 

.70 

.7586 

1.2552 

.6044 

1.70 

2.6456 

2.8288 

.9354 

72 

7838 

1.2706 

6169 

1.72 

2.7027 

2.8818 

9379 

74 

8094 

1.2865 

6291 

174 

2.7609 

2.9864 

9402 

76 

8353 

1.3030 

6411 

1.76 

2.8202 

2  9922 

9425 

78 

8615 

1.3199 

6527 

1.78 

2.8806 

3.0492 

9447 

.80 

.8881 

1.8374 

.6640 

1.80 

2.9422 

3.1075 

.9468 

82 

9150 

1.3555 

6751 

1.82 

3.0049 

8.1669 

9488 

84 

9428 

1.3740 

6858 

1.84 

3.0689 

3  2277 

9508 

86 

9700 

1  3982 

6963 

1.86 

3.1340 

3.2897 

9527 

88 

9981 

1.4128 

7064 

1.88 

3.2005 

3.3530 

9545 

.90 

1.0265 

1.4331 

.7163 

1.90 

3.2682 

3.4177 

.9562 

92 

1.0554 

1.4539 

7259 

1.92 

3.3372 

3.4838 

9579 

94 

1.0847 

1  4758 

7352 

1.94 

3.4075 

3.5512 

9595 

96 

1.1144 

1.4973 

7443 

1.96 

3.4792 

3.6201 

9611 

98 

1 . 1446 

1.5199 

7531 

1.98 

3.5523 

3.6904 

9626 

TABLES. 


163 


Table  I.     Hyperbolic  Functions. 


u. 

sinh  u. 

cosh  «. 

tanh  u. 

u. 

sinh  u. 

cosh  u. 

tanh  u. 

2.00 

3.6269 

3.7622 

.9640 

3.00 

10.0179 

10.0677 

.99505 

2.02 

3.7028 

3.8355 

9654 

3.02 

10.2212 

10.2700 

99524 

2.04 

3.7803 

3.9103 

9667 

3.04 

10.4287 

10.4765 

99543 

2.06 

3.8593 

3.9867 

9680 

3.06 

10.6403 

10.6872 

99561 

2.08 

3.9398 

4.0647 

9693 

3.08 

10.8562 

10.9022 

99578 

2.10 

4.0219 

4.1443 

.9705 

3.10 

11.0765 

11.1215 

.99594 

2.12 

4.1056 

4.2256 

9716 

3.12 

11.3011 

11.3453 

99610 

2  14 

4.1909 

4.3085 

9727 

3.14 

11.5303 

11.5736 

99626 

2.16 

4  2779 

4.3932 

9737 

3.16 

11.7641 

11.8065 

99640 

2.18 

4.8666 

4.4797 

9748 

3.18 

12.0026 

12.0442 

99654 

2.20 

4.4571 

4.5679 

.9757 

3.20 

12.2459 

12.2866 

.99668 

2.22 

4.5494 

4.6580 

9767 

3.22 

12.4941 

12.5340 

99681 

2.24 

4.6434 

4.7499 

9776 

3.24 

12.7473 

3  2.7864 

99693 

2.26 

4.7394 

4.8437 

9785 

3.26 

13  0056 

13.0440 

99705 

2.28 

4.8372 

4.9395 

9793 

3.28 

13.2691 

13.3067 

99717 

2.30 

4.9370 

5.0372 

.9801 

3.30 

13.5379 

13.5748 

.99728 

2.32 

5.0387 

5.1370 

9809 

3.32 

13.8121 

13.8483 

99738 

2.34 

5.1425 

5  2388 

9816 

3.34 

14.0918 

14.1273 

99749 

2.36 

5.2483 

5.3427 

9823 

3.36 

14.3772 

14.4120 

99758 

2.38 

5.3562 

5.4487 

9830 

3.38 

14.6684 

14.7024 

99768 

2.40 

5.4662 

5.5569 

.9837 

3.40 

14.9654 

14.9987 

.99777 

2.42 

5.5785 

5.6674 

9843 

3.42 

15.2684 

15.3011 

99786 

2.44 

5.6929 

5.7801 

9849 

8.44 

15.5774 

15.6095 

99794 

2.46 

5  8097 

5.8951 

9855 

3.46 

15.8928 

15.9242 

99802 

2.48 

5.9288 

6.0125 

9861 

3.48 

16.2144 

16.2453 

99810 

2.50 

6.0502 

6.1323 

.9866 

3  50 

16.5426 

16.5728 

.99817 

2  52 

6.1741 

6.2545 

9871 

8.52 

16.8774 

16.9070 

99824 

2.54 

6.3004 

6.3793 

9876 

8.54 

17.2190 

17.2480 

99831 

2.56 

6.4293 

6.5066 

9881 

8.56 

17.5674 

17.5958 

99838 

2.58 

6.5607 

6.6364 

9886 

8.58 

17.9228 

17.9507 

99844 

2.60 

6.6947 

6.7690 

.9890 

8.60 

18.2854 

18.3128 

.99850 

2.62 

6.8315 

6.9043 

9895 

3.62 

18.6554 

18.6822 

99856 

2.64 

6.9709 

7.0423 

9899 

3.64 

19.0328 

19.0590 

99862 

2.66 

7.1132 

7.1832 

9903 

3.66 

19.4178 

19.4435 

99867 

2.68 

7.2583 

7.3268 

9906 

3.68 

19.8106 

19.8358 

99872 

2.70 

7.4063 

7.4735 

.9910 

3.70 

20.2113 

20.2360 

.99877 

2.72 

7.5572 

7.6231 

9914 

8.72 

20.6201 

20.6443 

99882 

2.74 

7.7112 

7.7758 

9917 

3.74 

21.0371 

21.0609 

99887 

2.76 

7.8683 

7.9316 

9920 

3.76 

21.4626 

21.4859 

99891 

2.78 

8.0285 

8.0905 

9923 

3.78 

21.8966 

21.9194 

99896 

2.80 

8.1919 

8.2527 

.9926 

8.80 

22.3394 

22.3618 

.99900 

282 

8  3586 

8.4182 

9929 

3.82 

22.7911 

22.8131 

99904 

2.84 

8.5287 

8.5871 

9932 

3  84 

23.2520 

23.2735 

99907 

2.86 

8.7021 

8.7594 

9935 

3.86 

28.7221 

23.7432 

99911 

2  88 

8.8791 

8.9352 

9937 

3.88 

24.2018 

24.2224 

99915 

2  90 

9  0596 

9.1146 

.9940 

3.90 

24.6911 

24.7113 

.99918 

2.92 

9.2437 

9.2976 

9942 

3.92 

25.1903 

25.2101 

99«»21 

2.94 

9.4315 

9  4844 

9944 

8.94 

25.6996 

25.7190 

99924 

2.96 

9.6231 

9  6749 

9947 

8.96 

26  2191 

26.2382 

99927 

2.98 

9.8185 

9.8693 

9949 

3.98 

26.7492 

26.7679 

99930 

164 


HYPERBOLIC    FUNCTIONS.  [ChAP.  IV. 

Table  II.    Values  of  cosh  (x  -\-  iy)  and  sink  {x  +  iy). 


X  — 

o 

X  ■=. 

.1 

y 

a 

b 

c 

d 

a 

b 

c 

d 

0 

1.0000 

0000 

0000 

.0000 

1.0050 

.00000 

.10017 

.0000 

.1 

0.9950 

0998 

1.0000 

01000 

09967 

1000 

.3 

0.9801 

1987 

0.9850 

01990 

0SJ817 

1997 

.3 

0.9553 

2955 

0.9601 

02960 

09570 

2970 

.4 

.9211 

.3894 

.9357 

.03901 

.09226 

.3914 

.5 

8776 

4794 

8830 

04802 

08791 

4818 

.6 

8353 

5646 

8395 

05656 

08367 

5675 

.7 

7648 

6442 

7687 

06453 

07661 

6474 

.8 

.6967 

.7174 

.7002 

.07186 

.06979 

.7200 

.9 

6316 

7833 

6247 

07847 

06227 

7873 

1.0 

5403 

8415 

5430 

08439 

05412 

8457 

1.1 

4536 

8913 

4559 

08937 

04544 

8957 

1.2 

.3634 

.9330 

.3642 

.09336 

.03630 

0  9367 

1.3 

2675 

9636 

2688 

09652 

02680 

0.9684 

1.4 

1700 

9854 

1708 

09871 

01703 

0.9904 

1.5 

0707 

9975 

0711 

09992 

00709 

1.0025 

\Tt 

0000 

" 

<< 

1.0000 

0000 

10017 

00000 

1.0050 

X   = 

.4 

.= 

•5 

y 

a 

b 

c 

d 

a 

b 

c 

d 

0 

1.0811 

.0000 

.4108 

.0000 

1.1276 

.0000 

.5211 

.0000 

.1 

1.0756 

0410 

4087 

1079 

1.1230 

0520 

5185 

1126 

.2 

1.0595 

0816 

4026 

2148 

1.1051 

1025 

5107 

3340 

.3 

1.0328 

1314 

3924 

3195 

1.0773 

1540 

4978 

3333 

.4 

.9957 

.1600 

.3783 

.4310 

1.0386 

.2029 

.4800 

.4391 

.5 

9487 

1969 

3605 

5183 

0.9896 

2498 

4573 

5406 

.6 

8933 

2319 

3390 

6104 

0.9306 

2942 

4301 

6367 

.7 

8268 

2646 

3142 

6964 

0.8624 

3357 

3986 

7364 

.8 

.7533 

.2947 

.2862 

.7755 

.7856 

.3738 

.3631 

0.8089 

.9 

6730 

3318 

2553 

8468 

7009 

4082 

3239 

0.8833 

1.0 

5841 

3456 

2219 

9097 

6093 

4385 

2815 

0.9489 

1.1 

4904 

3661 

1863 

9635 

5115 

4644 

2364 

1.0050 

1.2 

.3917 

.3829 

.1488 

1.0076 

.4086 

.4857 

.1888 

1.0510 

1.3 

3893 

3958 

1099 

1.0417 

3016 

5031 

1394 

1.0865 

1.4 

1838 

4048 

0698 

1.0658 

1917 

5135 

0886 

1.1163 

1.5 

0765 

4097 

0291 

1.0784 

0798 

5198 

0369 

1.1348 

\Tt 

0000 

41 

08 

0000 

1.0811 

0000 

5211 

0000 

1.1276 

Art.  39.]  tables. 

Table  II.    Values  of  cosh  (x  +  iy)  and  sink  (x  +  iy). 


165 


X  = 

.2. 

X   = 

•3 

a 

b 

C 

d 

a 

b 

c 

d 

y 

1.0201 

.0000 

.2013 

.0000 

1.0453 

.0000 

.3045 

.0000 

0 

1.0150 

0201 

2003 

1018 

1.0401 

0304 

3030 

1044 

.1 

0.9997 

0400 

1973 

2027 

1.0245 

0605 

2985 

2077 

.2 

0.9745 

0595 

1923 

3014 

9987 

0900 

2909 

3089 

.3 

.9395 

.0784 

.1854 

.3972 

.9628 

.1186 

.2805 

.4071 

.4 

8952 

0965 

1767 

4890 

9174 

1460 

2672 

5012 

.5 

8419 

1137 

1662 

5760 

8627 

1719 

2513 

5903 

.6 

7802 

1297 

1540 

6571 

7995 

1962 

2329 

6734 

.7 

.7107 

.1444 

.1403 

.7318 

.7283 

.2184 

.2122 

.7498 

.8 

6341 

1577 

1252 

7990 

6498 

2385 

1893 

8188 

.9 

5511 

1694 

1088 

8584 

5648 

2562 

1645 

8796 

1.0 

4627 

1795 

0913 

9091 

4742 

2714 

1381 

9316 

1.1 

.8696 

.1877 

.0730 

0.9507 

.3788 

.2838 

.1103 

0.9743 

1.2 

2729 

1940 

0539 

0.9829 

2796 

2934 

0815 

1.0072 

1.8 

1734 

1984 

0342 

1.0052 

1777 

8001 

0518 

1.0301 

1.4 

0722 

2008 

0143 

1.0175 

0739 

3038 

0215 

1.0427 

1.5 

0000 

2013 

0000 

1.0201 

0000 

3045 

0000 

1.0453 

\it 

X  — 

.6 

X  = 

•7 

a 

b 

c 

d 

a 

b 

c 

d 

y 

1.1855 

.0000 

.6367 

.0000 

1.2552 

.0000 

.7586 

,0000 

0 

1.1795 

0636 

6335 

1183 

1.2489 

0757 

7548 

1253 

.1 

1.1618 

1265 

6240 

2355 

1.2301 

1542 

7435 

2404 

.2 

1 . 1325 

1881 

6082 

3503 

1.1991 

2242 

7247 

3709 

.3 

1.0918 

.2479 

.5864 

.4617 

1.1561 

.2954 

.6987 

.4888 

.4 

1.0403 

3052 

5587 

5684 

1.1015 

8637 

6657 

6018 

.5 

0.9784 

35&5 

5255 

6694 

1.0359 

4253 

6261 

7087 

.6 

0.9067 

4101 

4869 

7637 

0.9600 

4887 

5802 

8086 

.7 

.8259 

.4567 

.4436 

0.8504 

.8745 

.5442 

.5285 

0.9004 

.8 

7369 

4987 

3957 

0.9286 

7802 

5942 

4715 

0.9832 

.9 

6405 

5357 

3440 

0.9975 

6782 

6383 

4099 

1.0562 

1.0 

5377 

5674 

2888 

1.0565 

5693 

6760 

3441 

1.1186 

1.1 

.4296 

5934 

.2307 

1.1049 

.4548 

.7070 

.2749 

1.1699 

1.2 

3171 

6135 

1703 

1.1422 

3358 

7309 

2029 

1.2094 

1.3 

2015 

6274 

1082 

1 .1682 

2183 

7475 

1289 

1 . 2369 

1.4 

0839 

6351 

0450 

1.1825 

0888 

7567 

0537 

1.2520 

1.5 

0000 

6367 

0000 

1.1855 

0000 

7586 

0000 

1.2552 

\Tt 

16G 


HYPERBOLIC    FUNCTIONS.  [ChAP.  IV. 

Table  II.     Values  of  cosh  (x  -f-  iy)  and  sink  (x  +  ij^)- 


y 

X   = 

.8 

X  = 

•9 

a 

b 

c 

d 

a 

b 

c 

d 

0 

.1 

.2 
.3 

1.3374 

1.3308 
1.3108 
1.2776 

.0000 
0887 
1764 
2625 

.8881 
8837 
8704 
8484 

.0000 
1335 
2657 
3952 

1.4831 
1.4259 
1.4045 
1.3691 

.0000 
1025 
2039 
3034 

1.0265 
1.0214 
1.0061 
0.9807 

.0000 
1481 

2847 
4235 

.4 
.5 

.6 

.7 

1.2319 
1.1737 
1.1038 
1.0229 

.3458 
4258 
5015 
5721 

.8180 
7794 
7330 
6793 

.5208 
6412 
7552 
8616 

1.3200 
1.2577 
1.1828 
1.0961 

.3997 
4921 
5796 
6613 

.9455 

9008 
8472 
7851 

.5581 
6871 
8092 
9232 

.8 

.9 

1.0 

1.1 

.9318 
8314 
7226 
6067 

.6371 
6957 

7472 
7915 

.6188 
5521 
4798 
4028 

0.9595 
1.0476 
1.1254 
1.1919 

.9984 
8908 
7743 
6500 

.7364 
8041 
8638 
9148 

.7152 
6381 
5546 
4656 

1.0280 
1.1226 
1.2059 
1.2772 

1.2 
1.3 
1.4 
1.5 

.4846 
3578 
2273 
0946 

.8278 
8557 
8752 
8859 

.3218 
2376 
1510 
0628 

1.2465 

1.2887 
1.3180 
1.3341 

.5193 
3834 
2436 
1014 

0.9568 
0.9891 
1.0124 
1.0239 

.3720 
2746 
1745 
0726 

1.3357 
1  3809 
1.4122 
1.4295 

iTT 

0000 

.8881 

0000 

1.3374 

0000 

1.0265 

0000 

1.4331 

X  = 

1.2 

X   = 

1-3 

y 

a 

b 

C 

d 

a 

b 

c 

d 

0 

.1 

.2 

.3 

1.8107 
1.8016 
1.7746 
1.7298 

.0000 
1507 
2999 
4461 

1.5095 
1.5019 
1.4794 
1.4420 

.0000 
1808 
3598 
5351 

1.9709 
1.9611 
1.9316 
1.8829 

.0000 
1696 
3374 
5019 

1.6984 
1.6899 
1.6645 
1.6225 

.0000 
1968 
3916 
5824 

.4 
.5 
.6 

.7 

1.6677 
1.5890 
1.4944 
1.3849 

.5878 
7237 
8523 
9724 

1.3903 
1.3247 
1.2458 
1.1545 

0.7051 
0.8681 
1.0224 
1.1665 

1.8153 
1.7296 
1  6267 
1.5074 

.6614 

8142 

9590 

1.0941 

1.5643 
1.4905 
1.4017 
1.2990 

0.7675 
0.9449 
1.1131 
1.2697 

.8 

.9 

1.0 

1.1 

1.2615 
1.1255 
0.9783 
0.8213 

1.0828 
1.1824 
1.2702 
1.3452 

1.0517 
0.9383 
0.8156 
0.6847 

1.2989 
1.4183 
1.5236 
1.6137 

1.3731 
12251 
1.0649 
0.8940 

1.2183 
1.3304 
1.4291 
1.5136 

1.1833 
1.0557 
0.9176 
0.7704 

1.4139 
1.5439 
1.6585 
1.7565 

1.2 
1.3 
1.4 
1.5 

.6561 

4844 
3078 
1281 

1.4069 
1.4.544 

1.4875 
1.5057 

.5470 

4038 
2566 
1068 

1.6876 
1.7447 
1.7843 
1.8061 

.7142 
5272 
3350 
1394 

1.5830 
1.6365 
1.6737 
1.6941 

.6154 
4543 

2887 
1201 

1.8370 
1.8991 
1.9422 
1.9660 

\it 

0000 

1.5095 

0000 

1.8107 

0000 

1.6984 

0000 

1.9709 

Art.  39.]  tables. 

Table  II.     Values  of  cosh  {x  -\-  iy)  and  sink  {x  -\-  iy.) 


167 


X  = 

I.O 

X  — 

I.I 

a 

b 

1    c 

d 

a 

b 

c 

d 

y 

1.5431 
1.5354 
1.5123 
1.4743 

.0000 
1173 
3335 
3473 

1.1753 
1.1693 
1.1518 
1.1227 

.0000 
1541 
8066 
4560 

1.6685 
1.6602 
1.6353 
1.5940 

.0000 
1333 
2654 
3946 

1.3356 
1.8290 
1.8090 
1.2760 

.0000 
1666 
3:J15 
4931 

0 

.1 

.2 
.3 

1.4313 
1.3542 
1.2736 
1.1803 

.4576 
5634 
6686 
7571 

1.0824 
1.0314 
0.9699 

0.8988 

.6009 

7398 
8718 
9941 

1.5368 
1.4643 
1.3771 
1.2762 

.5201 
6408 
7542 

8604 

1.2302 
1.1721 
1.1024 
1.0216 

0  6498 
0.7999 
0.9421 
1.0749 

.4 
.5 
.6 

.7 

1.0751 
0.9593 
0.8337 
0.6999 

0.8430 
0.9206 
0.9889 
1.0473 

.8188 
7305 
6350 
5331 

1  1069 
1.2087 
1.2985 
1.3752 

1.1625 
1.0872 
0.9015 
0.7568 

0  9581 
1.0462 
1.1289 
1.1903 

.9306 

8302 
7217 
6058 

1  1969 

1.3070 
1.4040 
1.4870 

.8 

.9 

1.0 

1.1 

.5593 
4128 
2623 
1093 

1.0953 
1.1824 
1.1581 
1.1723 

.4258 
8144 
1998 
0831 

1.4382 
1.4869 
1.5218 
1.5392 

.6046 
4463 
2836 
1180 

1.2449 
1.2870 
1.3162 
1.3823 

.4840 
3o73 
2270 
0945 

1.5551 
1.6077 
1.6442 
1.6643 

1.2 
1.8 
1.4 
1.5 

0000 

1.1753 

0000 

15431 

0000 

1.3356 

0000 

1.6685 

\Tt 

x  = 

1-4 

X  ■= 

r.5- 

a 

b 

c 

d 

a 

b 

c 

d 

y 

3.1509 
3.1401 
3.1080 
3.0548 

.0000 
1901 
3783 
5628 

1.9043 
1.8948 
1.8663 
1.8192 

.0000 
2147 
4273 
6356 

3.3524 

2  3413 
2.3055 

3  2473 

.0000 
2126 
4230 
6292 

2.1293 

2.1187 
2.0868 
2.0842 

.0000 
2848 
4674 
6951 

0 

.1 

.2 
.3 

1.9811 
1.8876 
1.7753 
1.6451 

0.7416 
0.9130 
1.0753 
1.2368 

1.7540 
1.6712 
1.5713 
1.4565 

0.8376 
1.0312 
1.2145 
1.3856 

2.1667 
2.0644 
1.9415 
1.7992 

0.8292 
1.0208 
1.2023 
1.3717 

1.9612 

1.8686 
1.7574 
1.6286 

0.9161 
1.1278 
1.3288 
1.5155 

.4 
.5 
.6 

.7 

1.4985 
1.3370 
1.1623 
0.9756 

1.3661 
1.4917 
1.6034 
1.6971 

1.3268 
1.1888 
1.0289 
0.8638 

1.5480 
1.6849 
1.8099 
1.9168 

1.6389 
1.4623 
1.2710 
1.0671 

1.5275 
1.6679 
1.7917 
1.8976 

1.4835 
1.8286 
1.1505 
0.9659 

1.6875 
1.8427 
1.9795 
2.0965 

.8 

.9 

1.0 

1.1 

.7794 
5754 
3656 
1523 

1.7749 
1.8349 
1.8766 
1.8996 

.6900 
5094 
3237 
1347 

2.0047 
2.0725 
2.1196 
2.1455 

.8524 
6293 
3998 
1664 

1.9846 
2.0517 
2.0983 
2.1239 

.7716 
5696 
3619 
1506 

2.1925 
2.2667 

2.8182 
2.8465 

1.2 
1.3 
1.4 
1.5 

.0000 

1.9043 

0000 

2.1509 

.0000 

2.1293 

.0000 

2.3524 

\% 

168 


HYPERBOLIC    FUNCTIONS. 

Table  III. 


[Chap.  Y 


u 

gd  u 

6° 

u 

gd  u 

6° 

u 

gd  u 

e° 

00 

.0000 

0.000 

.60 

.5669 

32.483 

1.50 

1.1317 

64^843 

.02 

0200 

1.146 

.62 

5837 

38.444 

1.55 

1.1525 

66.034 

.04 

04U0 

2.291 

.64 

6003 

34.395 

1.60 

1.1724 

67.171 

.06 

0600 

3.436 

.66 

6167 

35  336 

1.65 

1.1913 

68.257 

.08 

0799 

0.579 

.68 

6329 

36.265 

1.70 

1.2094 

69.294 

.10 

.0998 

5.720 

.70 

.6489 

37.183 

1.75 

1.2267 

70.284 

.12 

1197 

6.859 

.72 

6648 

38.091 

1.80 

1.2432 

71.228 

.14 

1395 

7.995 

.74 

6804 

38.987 

1.85 

1.2589 

72.128 

.16 

1593 

9.128 

.76 

6958 

39.872 

1.90 

1.2739 

72.987 

.18 

1790 
.1987 

10.258 
11.384 

.78 
.80 

7111 
.7261 

40.746 
41.608 

1.95 

1.2881 
1.3017 

73.805 

.30 

2.00 

74.584 

.22 

2183 

12.505 

.82 

7410 

42.460 

2.10 

1.3271 

76.037 

.24 

2377 

13.621 

.84 

7557 

43.299 

2.20 

1.3501 

77.354 

.26 

2571 

14.732 

.86 

7702 

44.128 

2.30 

1.3710 

78.549 

.28 

2764 

15.837 

.88 

7844 

44.944 

2.40 

1.3899 

79.633. 

.30 

.2956 

16.937 

.90 

.7985 

45.750 

2.50 

1.4070 

80.615 

.32 

3147 

18.030 

.92 

8123 

46.544 

2  60 

1.4227 

81.513 

.34 

3336 

19.116 

.94 

8260 

47.326 

2.70 

1.4366 

82.310 

.36 

3525 

20.195 

.96 

8394 

48.097 

2.80 

1.4493 

83.040 

.38 

3712 
.3897 

21.267 
22.331 

.98 

8528 
.8658 

48.857 
49.605 

2.90 
3  00 

1.4609 
1.4713 

83.707 

.40 

1.00 

84.301 

.42 

4082 

23.386 

1.05 

8976 

51.428 

3.10 

1.4808 

84.841 

.44 

4264 

24.434 

1.10 

9281 

53.178 

3.20 

1.4894 

85.336 

.46 

4446 

25.473 

1.15 

9575 

54  860 

3.30 

1.4971 

85.775 

.48 

4626 

26.503 

1.20 

9857 

56476 

3.40 

1.5041 

H6.177 

.50 

.4804 

27.524 

1.25 

1.0127 

58.026 

3.50 

1.5104 

86.541 

.52 

4980 

28.535 

1.30 

1.0387 

59.511  ■ 

3.60 

1.5162 

86.870 

.54 

5155 

29.537 

1.35 

1.0635 

60.933 

3.70 

1.5214 

87  168 

.56 

5328 

30.529 

1.40 

1.0873 

62.295 

3.80 

1.5261 

87.437 

.58 

5500 

31.511 

1.45 

1. 1100 

63.598 

3.90 

1.5303 

87.681 

Table  IV. 


u 

gd  u 

log  sinh  u 

log  cosh  u 

u 

gd  u 

log  sinh  u 

log  cosh  u 

4.0 

1.5342 

1.4360 

1.4363 

5.5 

1.5626 

2.08758 

2.08760 

4.1 

1.5377 

1.4795 

1.4797 

5.6 

1.5634 

2.13101 

2.13103 

4.2 

1.5408 

1.5229 

1.5231 

5.7 

1.5641 

2.17444 

2.17445 

4.3 

1.5437 

1.5664 

1.5665 

5.8 

1.5648 

2.21787 

2.21788 

4.4 

1.5463 
1.5486 

1.6098 
1.6532 

1.6099 
1.6533 

5.9 

1.5653 
1  5658 

2.26130 
2.30473 

3.26131 

4.5 

6.0 

3.30474 

4.6 

1.5507 

1.6967 

1.6968 

6.2 

1.5667 

2.39159 

3.39160 

4.7 

1  5526 

1.7401 

1.7403 

6.4 

1.5675 

2.47845 

3.47846 

4.8 

1.5543 

1.7836 

1.7836 

6.6 

1.5681 

2.56531 

3.56531 

4.9 

1.5559 
15573 

1.8270 

1.8704 

1.8370 
1.8705 

6.8 

1.5686 
1.5690 

2.65217 
2.73903 

3.65217 

5.0 

7.0 

3.73903 

5.1 

1.5586 

1.9139 

1.9139 

7.5 

1.5697 

2.95618 

3.95618 

5.2 

1.5598 

1.9573 

1.9573 

8.0 

1.5701 

3.17333 

3.17333 

5.3 

1.5608 

2.0007 

3.0007 

8.5 

1.5704 

3.39047 

3.39047 

5.4 

1.5618 

3.0443 

2.0442 

9.0 

1.5705 

3.60762 

3.60762 

00 

1.5708 

00 

00 

Art.  1,]  HISTORY    AND   DESCRIPTION.  169 


Chapter  V. 

HARMONIC  FUNCTIONS. 

By  William    E.  Byerly, 

Professor  of  Mathematics  in  Harvard  University. 

Art.  1.     History  and  Description. 

What  is  known  as  the  Harmonic  Analysis  owed  its  origin 
and  development  to  the  study  of  concrete  problems  in  various 
branches  of  Mathematical  Physics,  which  however  all  involved 
the  treatment  of  partial  differential  equations  of  the  same 
general  form. 

The  use  of  Trigonometric  Series  was  first  suggested  by 
Daniel  Bernouilli  in  1753  in  his  researches  on  the  musical 
vibrations  of  stretched  elastic  strings,  although  Bessel's  Func- 
tions had  been  already  (1732)  employed  by  him  and  by  Euler 
in  dealing  with  the  vibrations  of  a  heavy  string  suspended  from 
one  end;  and  Zonal  and  Spherical  Harmonics  were  introduced 
by  Legendre  and  Laplace  in  1782  in  deaHng  with  the  attrac- 
tion of  solids  of  revolution. 

The  analysis  was  greatly  advanced  by  Fourier  in  1812-1824 
in  his  remarkable  work  on  the  Conduction  of  Heat,  and  im- 
portant additions  have  been  made  by  Lam^  (1839)  ^^^  ^y  a 
host  of  modern  investigators. 

The  differential  equations  treated  in  the  problems  which 
have  just  been  enumerated  are 

g=^.'B  (I) 


170  HARMONIC    FUNCTIONS.  [Chap.  V. 

for  the  transverse  vibrations  of  a  musical  string ; 

for  small  transverse  vibrations  of  a  uniform  heavy  string  sus- 
pended from  one  end ; 

which  is  Laplace's  equation  ;  and 

for  the  conduction  of  heat  in  a  homogeneous  solid. 

Of  these  Laplace's  equation  (5),  and  (4)  of  which  (3)  is  a 
special  case,  are  by  far  the  most  important,  and  we  shall  con- 
cern ourselves  mainly  with  them  in  this  chapter.  As  to  their 
interest  to  engineers  and  physicists  we  quote  from  an  article 
in  The  Electrician  of  Jan.  26,  1894,  by  Professor  John  Perry: 

"  There  is  a  well-known  partial  differential  equation,  which  is 
the  same  in  problems  on  heat-conduction,  motion  of  fluids,  the 
establishment  of  electrostatic  or  electromagnetic  potential,  certain 
motions  of  viscous  fluid,  certain  kinds  of  strain  and  stress,  currents 
in  a  conductor,  vibrations  of  elastic  solids,  vibrations  of  flexible 
strings  or  elastic  membranes,  and  innumerable  other  phenomena. 
The  equation  has  always  to  be  solved  subject  to  certain  boundary 
or  limiting  conditions,  sometimes  as  to  space  and  time,  sometimes 
as  to  space  alone,  and  we  know  that  if  we  obtain  any  solution  of  a 
particular  problem,  then  that  is  the  true  and  only  solution.  Further- 
more, if  a  solution,  say,  of  a  heat-conduction  problem  is  obtained 
by  any  person,  that  answer  is  at  once  applicable  to  analogous  prob- 
lems in  all  the  other  departments  of  physics.  Thus,  if  Lord  Kel- 
vin draws  for  us  the  lines  of  flow  in  a  simple  vortex,  he  has  drawn 
for  us  the  lines  of  magnetic  force  about  a  circular  current:  if 
Lord  Rayleigh  calculates  for  us  the  resistance  of  the  mouth  of  an 
organ-pipe,  he  has  also  determined  the  end  effect  of  a  bar  of  iron 
which  is  magnetized;  when  Mr.  Oliver  Heaviside  shows  his  match- 


Art.  1.]  HISTORY    AND    DESCRIPTION.  171 

less  skill  and  familiarity  with  Bessel's  functions  in  solving  electro- 
magnetic problems,  he  is  solving  problems  in  heat-conductivity  or 
the  strains  in  prismatic  shafts.  How  difficult  it  is  to  express  exactly 
the  distribution  of  strain  in  a  twisted  square  shaft,  for  example,  and 
yet  how  easy  it  is  to  understand  thoroughly  when  one  knows  the 
perfect-fluid  analogy!  How  easy,  again,  it  is  to  imagine  the  electric 
current  density  everywhere  in  a  conductor  when  transmitting  alter- 
nating currents  when  we  know  Mr.  Heaviside's  viscous-fluid  analogy, 
or  even  the  heat-conduction  analogy! 

"  Much  has  been  written  about  the  correlation  of  the  physical 
sciences;  but  when  we  observe  how  a  young  man  who  has  worked 
almost  altogether  at  heat  problems  suddenly  shows  himself  ac- 
quainted with  the  most  difficult  investigations  in  other  departments 
of  physics,  we  may  say  that  the  true  correlation  of  the  physical 
sciences  lies  in  the  equation  of  continuity 


In  the  Theory  of  the  Potential  Function  in  the  Attraction 
of  Gravitation,  and  in  Electrostatics  and  Electrodynamics,* 
Fin  Laplace's  equation  (3)  is  the  value  of  the  Potential  Func- 
tion, at  any  external  point  (x,  y,  z)^  due  to  any  distribution  of 
matter  or  of  electricity ;  in  the  theory  of  the  Conduction  of 
Heat  in  a  homogeneous  solid  f  F  is  the  temperature  at  any 
point  in  the  solid  after  the  stationary  temperatures  have  been 
established,  and  in  the  theory  of  the  irrotational  flow  of  an 
incompressible  fluid  %  V  \s  the  Velocity  Potential  Function 
and  (3)  is  known  as  the  equation  of  continuity. 

If  we  use  spherical  coordinates,  (3)  takes  the  form 


a  sin  e 


ira'(rF)      I    n— aW  ,     I    a^n^Q.     ,,. 

?L      dr'     ■^sin6/  dB  ^sin'^a0^J         '        ^^> 

*  See  Peirce's  Newtonian  Potential  Function.     Boston. 

f  See  Fourier's  Analytic  Theory  of  Heat.     London  and  New  York,  1878  ; 
or  Riemann's  Partielle  Differentialgleichungen.      Brunswick. 

X  See  Lamb's  Hydrodynamics.     London  and  New  York,  1895. 


172  HARMONIC    FUNCTIONS.  [ChAP.  V. 

and  if  we  use  cylindrical  coordinates,  the  form 

In  the  theory  of  the  Conduction  of  Heat  in  a  homogene- 
ous solid,*  u  in  equation  (4)  is  the  temperature  of  any  point 
{x,  y,  z)  of  the  solid  at  any  time  /,  and  a"  is  a  constant  deter- 
mined by  experiment  and  depending  on  the  conductivity  and 
the  thermal  capacity  of  the  solid. 

Art.  2.    Homogeneous  Linear  Differential  Equations. 

The  general  solution  of  a  differential  equation  is  the  equa- 
tion expressing  the  most  general  relation  between  the  primi- 
tive variables  which  is  consistent  with  the  given  differential 
equation  and  which  does  not  involve  differentials  or  derivatives. 
A  general  solution  will  always  contain  arbitrary  (i.e.,  undeter- 
mined) constants  or  arbitrary  functions. 

A  particular  solution  of  a  differential  equation  is  a  relation 
between  the  primitive  variables  which  is  consistent  with  the 
given  differential  equation,  but  which  is  less  general  than  the 
general  solution,  although  included  in  it. 

Theoretically,  every  particular  solution  can  be  obtained 
from  the  general  solution  by  substituting  in  the  general  solu- 
tion particular  values  for  the  arbitrary  constants  or  particular 
functions  for  the  arbitrary  functions;  but  in  practice  it  is  often 
easy  to  obtain  particular  solutions  directly  from  the  differential 
equation  when  it  would  be  difficult  or  impossible  to  obtain  the 
general  solution. 

{a)  If  a  problem  requiring  for  its  solution  the  solving  of  a 
differential  equation  is  determinate,  there  must  always  be  given 
in  addition  to  the  differential  equation  enough  outside  condi- 
tions for  the  determination  of  all  the  arbitrary  constants  or 
arbitrary  functions  that  enter  into  the  general  solution  of  the 
equation  ;  and  in  dealing  with  such  a  problem,  if  the  differen- 
tial equation  can  be  readily  solved  the  natural  method  of  pro- 


Art.  3.]    homogeneous  linear  differential  equations.         175 

cedure  is  to  obtain  its  general  solution,  and  then  to  determine 
the  constants  or  functions  by  the  aid  of  the  given  conditions. 

It  often  happens,  however,  that  the  general  solution  of  the 
differential  equation  in  question  cannot  be  obtained,  and  then,. 
since  the  problem,  if  determinate,  will  be  solved,  if  by  any 
means  a  solution  of  the  equation  can  be  found  which  will  also 
satisfy  the  given  outside  conditions,  it  is  worth  while  to  try  to 
get  particular  solutions  and  so  to  combine  them  as  to  form  a 
result  which  shall  satisfy  the  given  conditions  without  ceasing 
to  satisfy  the  differential  equation. 

(b)  A  differential  equation  is  linear  when  it  would  be  of  the 
first  degree  if  the  dependent  variable  and  all  its  derivatives 
were  regarded  as  algebraic  unknown  quantities.  If  it  is  linear 
and  contains  no  term  which  does  not  involve  the  dependent 
variable  or  one  of  its  derivatives,  it  is  said  to  be  linear  and 
homogeneous. 

All  the  differential  equations  given  in  Art.  i  are  linear  and 
homogeneous. 

{c)  If  a  value  of  the  dependent  variable  has  been  found 
which  satisfies  a  given  homogeneous,  linear,  differential  equa- 
tion, the  product  formed  by  multiplying  this  value  by  any 
constant  will  also  be  a  value  of  the  dependent  variable  which 
will  satisfy  the  equation. 

For  if  all  the  terms  of  the  given  equation  are  transposed 
to  the  first  member,  the  substitution  of  the  first-named  value 
must  reduce  that  member  to  zero;  substituting  the  second 
value  is  equivalent  to  multiplying  each  term  of  the  result  of 
the  first  substitution  by  the  same  constant  factor,  which  there- 
fore may  be  taken  out  as  a  factor  of  the  whole  first  member. 
The  remaining  factor  being  zero,  the  product  is  zero  and  the 
equation  is  satisfied. 

{d)  If  several  values  of  the  dependent  variable  have  been 
found  each  of  which  satisfies  the  given  differential  equation, 
their  sum  will  satisfy  the  equation  ;  for  if  the  sum  of  the  values 
in  question  is  substituted  in  the  equation,  each  term  of  the  sum 


174  HARMONIC    FUNCTIONS.  [ChaP.  V. 

will  give  rise  to  a  set  of  terms  which  must  be  equal  to  zero,  and 
therefore  the  sum  of  these  sets  must  be  zero. 

{e)  It  is  generally  possible  to  get  by  some  simple  device 
particular  solutions  of  such  differential  equations  as  those  we 
have  collected  in  Art.  i.  The  object  of  this  chapter  is  to  find 
methods  of  so  combining  these  particular  solutions  as  to  satisfy 
any  given  conditions  which  are  consistent  with  the  nature  of 
the  problem  in  question. 

This  often  requires  us  to  be  able  to  develop  any  given  func- 
tion of  the  variables  which  enter  into  the  expression  of  these 
conditions  in  terms  of  normal  forms  suited  to  the  probkm  with 
which  we  happen  to  be  dealing,  and  suggested  by  the  form  of 
particular  solution  that  we  are  able  to  obtain  for  the  differential 
equation. 

These  normal  forms  are  frequently  sines  and  cosines,  but 
they  are  often  much  more  complicated  functions  known  as 
Legendre's  Coefificients,  or  Zonal  Harmonics ;  Laplace's  Coef- 
ficients, or  Spherical  Harmonics ;  Bessel's  Functions,  or  Cylin- 
drical Harmonics;  Lame's  Functions,  or ,  Ellipsoidal  Har- 
monics; etc. 

Art.  3.    Problem  in  Trigonometric  Series. 

As  an  illustration  let  us  consider  the  following  problem : 
A  large  iron  plate  7t  centimeters  thick  is  heated  throughout 
to  a  uniform  temperature  of  lOO  degrees  centigrade;  its  faces 
are  then  suddenly  cooled  to  the  temperature  zero  and  are  kept 
at  that  temperature  for  5  seconds.  What  will  be  the  tempera- 
ture of  a  point  in  the  middle  of  the  plate  at  the  end  of  that 
time?     Given  a""  =0.185  in  C.G.S.  units. 

Take  the  origin  of  coordinates  in  one  face  of  the  plate 
and  the  axis  of  X  perpendicular  to  that  face,  and  let  u  be  the 
temperature  of  any  point  in  the  plate  f  seconds  after  the  cool- 
ing begins. 

We  shall  suppose  the  flow  of  heat  to  be  directly  across  the 
plate  so  that  at  any  given  time  all  points  in  any  plane  parallel 


Art.  3.]  PROBLEM    IN    TRIGONOMETRIC    SERIES.  175 

to   the   faces   of   the   plate   will   have   the   same   temperature. 
Then  u  depends  upon  a  single  space-coordinate  x  ;  —  =  o  and 

—  =  O,  and  (4),  Art.  I,  reduces  to 

Obviously,  u  =  ioo°  when  /  =  o,  (2) 

u  =i       o  when  x  ^=  o,  (3) 

and  u  =       o  when  x  =  7t;  (4) 

and  we  need  to  find  a  solution  of  (i)  which  satisfies  the  con- 
ditions (2),  (3),  and  (4). 

We  shall  begin  by  getting  a  particular  solution  of  (i),  and 
we  shall  use  a  device  which  always  succeeds  when  the  equa- 
tion is  linear  and  homogeneous  and  has  constant  coefficients. 

Assume*  u  =  e^^'+y^  where  /3  and  y  are  constants;  substi- 
tute in  (i)  and  divide  through  by  ^P*+y'  and  we  get  y  =  a^/3^ ; 
and  if  this  condition  is  satisfied,  ii  =  r^^+v'  is  a  solution  of  (i). 

u  =  ^*+«'^'^  is  then  a  solution  of  (1)  no  matter  what  the 
value  of  /3. 

We  can  modify  the  form  of  this  solution  with  advantage. 
Let  /3  =  pii,f  then  u  =  ^-«V'Vm^»  is  a  solution  of  (i),  as  is  also 

By  (^),  Art.  2, 

is  a  solution,  as  is  also 

u  =  ^-v^^^f £ /  =  ^-«V'^  sin  fAX  ;  (6) 

and  ^  is  entirely  arbitrary. 

*  This  assumption  must  be  regarded  as  purely  tentative.     It  must  be  tested 
by  substituting  in  the  equation,  and  is  justified  if  it  leads  to  a  solution. 
f  The  letter  i  will  be  used  to  represent  V  —  i. 


176  HARMONIC    FUNCTIONS.  [ChAP.  V. 

By  giving  different  values  to  ^  we  get  different  particular 
solutions  of  (i) ;  let  us  try  to  so  combine  them  as  to  satisfy  our 
conditions  while  continuing  to  satisfy  equation  (i). 

u  =  ^-*V'^^  sin  ^x  is  zero  when  ;i;  =  o  for  all  values  oi  ju;  it 
is  zero  when  ;i:  =  ;r  if  yw  is  a  whole  number.  If,  then,  we  write 
u  equal  to  a  sum  of  terms  of  the  form  Ae'''"'*'''^  sin  mx,  where 
m  is  an  integer,  we  shall  have  a  solution  of  (i)  (see  {d),  Art.  2) 
which  satisfies  (3)  and  (4). 

Let  this  solution  be 

u  =  A.e-'"'^  sin  x  +  A,e-'^'^  sin  2x  +  A.e-^'"'^  sin  3;r  +  ...,  (7) 

A^y  A^,  A^,  .  .  .  being  undetermined  constants. 
When  /  =  o,  (7)  reduces  to 

u  =  A^  sin  X  -\-  A^  sin  2;r  +  ^3  sin  34;  +  .  .  .  .  (8) 

If  now  it  is  possible  to  develop  unity  into  a  series  of  the 
form  (8)  we  have  only  to  substitute  the  coefficients  of  that 
series  each  multiplied  by  100  for  A^,  A^y  A^»  .  .in  (7)  to  have 
a  solution  satisfying  (i)  and  all  the  equations  of  condition  (2), 
(3),  and  (4). 

We  shall  prove  later  (see  Art.  6)  that 

I  =  —    sin  :r  +  "  sin  3;ir  +  ~  sin  5;i;  +  •  •  • 
for  all  values  of  x  between  o  and  7t.     Hence  our  solution  is 
u  =  ^r^'""'  sin  X  +  -e-''^''  sin  sx  +  -e-^^''  sin  5;^  +  .  . .  |  (9) 

To  get  the  answer  of  the  numerical  problem  we  have  only 

to  compute  the  value  of  u  when  x  =  —  and  /  =  5  seconds.     As 

there  is  no  object  in  going  beyond  tenths  of  a  degree,  four- 
place  tables  will  more  than  suffice,  and  no  term  of  (9)  beyond 

7t 

the  first  will  affect  the  result.     Since  sin   -  =  I,  we  have  to 

2 

compute  the  numerical  value  of 


Art.  4.]  problem  in  zonal  harmonics.  177 


- — ^-'*'"    where 
n 

a'  = 

=  0.185     and     t=S. 

log  a^    =  9.2672  — 

10 

log  400     =  2.6061 

log  /      =  0.6990 

colog  TT    =  9.5059  — 

10 

log  ^V  =  9.9662  — 

10 

colog  ^''  =  9.5982  - 

10 

log  log  e      =  9-6378  — 

10 

log  71         =  1. 7 102 

log  log  ^'^  =  9.6040  — 

10 

0                     / 

log  e^'^  =0.4018 

u               =5i°.3. 

If  the  breadth  of  the  plate  had  been  c  centimeters  instead 
of  Tt  centimeters  it  is  easy  to  see  that  we  should  have  needed 
the  development  of  unity  in  a  series  of  the  form 

^       .      TtX    ,       ^       .      271X    ,       ^       .      Xnx    , 

A^  sin  —  +  ^,  sm  — -  +  A,  sm  -—  +  .... 

Prob.  I.  An  iron  slab  50  centimeters  thick  is  heated  to  the  tem- 
perature 100  degrees  Centigrade  throughout.  The  faces  are  then  sud- 
denly cooled  to  zero  degrees,  and  are  kept  at  that  temperature  for 
10  minutes.  Find  the  temperature  of  a  point  in  the  middle  of  the 
slab,  and  of  a  point  10  centimeters  from  a  face  at  the  end  of  that 
time.  Assume  that 

4/    .       7tX     ^      T      .       -KTZX     ,     I      .       KTtX   ,  \    ^ 

I  =  -  Sin \-  -  sin h"  sin h...    from  x  =  o  to  x  =  c. 

Ans.  84°.o;  49°^4. 

Art.  4.    Problem  in  Zonal  Harmonics. 

As  a  second  example  let  us  consider  the  following  problem  : 
Two  equal  thin  hemispherical  shells  of  radius  unity  placed 
together  to  form  a  spherical  surface  are  separated  by  a  thin 
layer  of  air.  A  charge  of  statical  electricity  is  placed  upon 
one  hemisphere  and  the  other  hemisphere  is  connected  with 
the  ground,  the  first  hemisphere  is  then  found  to  be  at  poten- 
tial I,  the  other  hemisphere  being  of  course  at  potential  zero. 
At  what  potential  is  any  point  in  the  "  field  of  force"  due  to 
the  charge  ? 

We  shall  use  spherical  coordinates  and  shall  let  Fbe  the 
potential  required.     Then  Fmust  satisfy  equation  (5),  Art.  r. 


178  HARMONIC    FUNCTIONS.  [ChaP.  V. 

But  since  from  the  symmetry  of  the  problem  V  is  obviously 
independent  of  0,  if  we  take  the  diameter  perpendicular  to  the 

plane  separating  the  two  conductors  as  our  polar  axis,  — —  is 
zero,  and  our  equation  reduces  to 

dr'      ^  sin  d  dd  ^./ 

Fis  given  on  the  surface  of  our  sphere,  hence 

V  =  /{0)     when     r=i,  (2) 

where y(6^)  =  i  if  o  <  ^  <  -,  and  /(O)  =  o  li  -  <  6  <  tt, 

2  2 

Equation  (2)  and  the  implied  conditions  that  V  is  zero  at 
an  infinite  distance  and  is  nowhere  infinite  are  our  conditions. 

To  find  particular  solutions  of  (i)  we  shall  use  a  method 
which  is  generally  effective.  Assume*  that  V=R&  where  R 
is  a  function  of  r  but  not  of  0,  and  Q  is  a  function  of  6  but 
not  of  r.     Substitute  in  (i)  and  reduce,  and  we  get 

I  rd\rR)  ^ I       4^^"  ^^e).  (3) 

R     dr'  Q  sin  6         dd 

Since  the  first  member  of  (3)  does  not  contain  B  and  the 
second  does  not  contain  r  and  the  two  members  are  identically 
equal,  each  must  be  equal  to  a  constant.  Let  us  call  this 
constant,  which  is  wholly  undetermined,  m{m-\-  i) ;  then 

,  .     ^.  <3'lsin   d-jjr] 

r  dHrR)  i  ^         dO  I  /      ,    ,x 

—  -^ — ^  = =  mim -4-  I) ; 

R     dr'  esmO  dO  ^     ^    >" 

whence  ^~dr^  ~  ^^^  -\- \)R  =  o,  {4) 

/         dS\ 

^"^  ^Te — de —  +  M^+i)0-o.  (5) 

*  See  the  first  foot-note  on  page  175. 


Art.  4.]  problem  in  zonal  harmonics.  179 

Equation  (4)  can  be  expanded  into 

and    can    be   solved    by  elementary   methods.     Its   complete 
solution  is 

R  =^  Ar"^ -\- Br-'^'-K  (6) 

Equation  (5)  can  be  simplified  by  changing  the  independ- 
ent variable  to  x  where  x  =  cos  0.     It  becomes 

d  r  d&  1 

an  equation  which  has  been  much  studied  and  which  is  known 
as  Legendre's  Equation. 

We  shall  restrict  m,  which  is  wholly  undetermined,  to  posi- 
tive whole  values,  and  we  can  then  get  particular  solutions  of 
(7)  by  the  following  device : 

Assume*  that  Q  can  be  expressed  as  a  sum  or  a  series  of 
terms  involving  whole  powers  of  x  multiplied  by  constant 
coefficients. 

Let  O  =  ^a^x""  and  substitute  in  (7).     We  get 

^n{n  —  i)a„x*'~''  —  n{n  +  O^w-^**  +  fnini  +  i)^^^'']  =  o,     (8) 

where  the  symbol  ^  indicates  that  we  are    to    form    all  the 
terms  we  can  by  taking  successive  whole  numbers  for  ;/. 

Since  (8)  must  be  true  no  matter  what  the  value  of  x,  the 
coefficient  of  any  given  power  of  x^  as  for  instance  x^,  must 
vanish.     Hence 

{k  +  2){k  +  i>,+,  -  k{k  +  i)^^  +  M^  +  0«*  =  o, 

ot.^  ^  minimi) -k{k-\-i) 

and  a,,^  = ^________^^.  (9) 

If  now  any  set  of  coefficients  satisfying  the  relation  (9)  be 
taken,  Q  =  ^a^^  will  be  a  solution  of  (7). 

If  k  =z  m^  then  Uk^^  =  o,     a^^^  =  o,    etc. 

*  See  the  first  foot-note  on  page  175. 


180  HARMONIC    FUNCTIONS.  [Chap,  V. 

Since  it  will  answer  our  purpose  if  we  pick  out  the  simplest 
set  of  coefificients  that  will  obey  the  condition  (9),  we  can  take 
a  set  including  a,„. 

Let  us  rewrite  (9)  in  the  form 

_  (^  +  2)(^+l) 

^*  -       {ni-  k\ni  -\-k~  I)'  ^^°> 

We  get  from  (10),  beginning  with  k  =1  m  —  2, 

_  m{in  —  i) 

'^"-'  ~  ~  2,  {2m-  I)    "' 

_  mjni  —  i){m  —  2){m  —  3) 
"-""    2.4,{2m-i){2m-z)'"' 

_        m(m  —  i){m  —  2){m  —  '^){m  —  4)(w  —  5) 
^'^-«  ""  ~         2.4.6.  (2W  -  i){2m  -  i){2M  -  5)      ^'"^  ^*^' 

If  m  is  even  we  see  that  the  set  will  end  with  a^\  if  m 
is  odd,  with  a^. 

^    ■        V  mim  —  i)    «  o 

*"  L  2A2m—i) 


m{m  -  i){m  -  2)m  -  3)^^. 


-...]. 


2.4.  {2m  —  i){2m  —  3) 

where  a^  is  entirely  arbitrary,  is,  then,  a  solution  of  (7).     It  is 
found  convenient  to  take  a^  equal  to 

(2;;/  —  i){2m  —  3) ...  I 

and  it  will  be  shown  later  that  with  this  value  of  ^^ ,  0  —  i 
when  X  =^  I, 

0  is  a  function  of  x  and  contains  no  higher  powers  of  x 
than  x"*".     It  is  usual  to  write  it  as  PJx). 

We  proceed  to  write  out  a  few  values  of  P„t(^)  from  the 
formula 

P  (^\  -  (2^  -  i){2m  -  3)  ♦  >  ♦  I  r^^  __  nt{m-  i) 

^-W  -  ^I  [_^  2.(2W-lf 


2.  {2m—  l) 

2.4.  {2m  —  i){2m  —  3) 


m{m  -  i){m  -  2)(m  -  3)  ^^_,  _  _  1  („) 


h(l2) 


Art.  4.]  PROBLEM    IN    ZONAL    HARMONICS.  181 

We  have : 

Fix)  =  I  or  P,(cos  e)=i,  ~~ 

Pipe)  =.  X  or   P,(cos  B)  =  cos  6, 

Plx)  =:  iisx'  -  I)     or   P,{cos  6)  =  i(3  cos^^  -  i), 
PM  =  i(5^'  -  3-^)   or   P3(cos  e)  =  4(5  cos^^  -  3  cos  0), 
Pk^)  =  i(35-^'  -  SO^r'  +  3)  or 

P,(cos  0)  =  i(35  cos*^  -  30  cos^^  +  3), 

P.{^)  =  i(p3^'  -  70^'  +  1 5-^)  or 

P,{cos  6)  ^  4(63  cos'  8  —  JO  cos''<y+  15  cos  ^).  ^ 

We  have  obtained  ©  =  P,n{^)  as  a  particular  solution  of  (7), 
and  0  =  P,„(cos  0)  as  a  particular  solution  of  (5).  /^„/-r)  or 
P,„(cos  6)  is  a  new  function,  known  as  a  Legendre's  Coefficient, 
or  as  a  Surface  Zonal  Harnionic,  and  occurs  as  a  normal  form 
in  many  important  problems. 

y—^  r'^pj^Qos  6)  is  a  particular  solution  of  (i),  and  r*^P^{cos  0) 
is  sometimes  called  a  Solid  Zonal  Harmonic. 

V  =  A,P,{cos  6)  +  y4,rPAcos  6)  +  AyP,{cos  6) 

-\-Aypicose)-^,,.   (13) 

satisfies  (i),  is  not  infinite  at  any  point  within  the  sphere,  and 

reduces  to 

F=  A,Plcos  6)  +  A^P^cos  d)  +  ^,P,(cos  6) 

+  ^,P3(cos^)+...     (14) 
when  r  =  I. 

A,P,(cos  6)  ,  A.PXcose)    .  A,P,{cos  6) 


V  = 


r 


+  --^p -  +  '"    (15) 

satisfies  (i),  is  not  infinite  at  any  point  without  the  sphere,  is 
equal  to  zero  when  r  =  co ,  and  reduces  to  (14)  when  r  =  i. 

If  then  we  can  develop /(^)  [see  eq.  (2)]  into  a  series  of  the 
form  (14),  we  have  only  to  put  the  coefficients  of  this  series  in 
place  of  the  A^,  A^,  A^,  ...  in  (13)  to  get  the  value  of  Ffor  a 
point  within  the  sphere,  and  in  (15)  to  get  the  value  of  Fat  a 
point  without  the  sphere. 


182  HARMONIC    FUNCTIONS.  [Chap.  V. 

We  shall  see  later  (Art.  i6,  Prob.  22)  that  if  f{0)  =  i  for 
o  <  6*  <  —  and  f{d)  =  o  for  —  <  6^  <  tt, 

f^e)  =  \+\pxcose)^l'\' Pilose) 

+  — -'-^Picos^)-...      (i6) 

'      12      2.4    *^  ^  ^      ^ 

Hence  our  required  solution  is 


+  — •  ^—r'Plco^  ^-  . . .     (17) 
~  12    2.4      '^         ^  ^  ^^ 


at  an  internal  point ;  and 


V  =  —  +  -  -^ Picose)^'^-'-\ PlcosB) 


1  .Li.ll3ip(cos^)-...     (18) 


?t  an  external  point. 

If  r  =  -  and  6  =  o,  (17)  reduces  to 

X.      I    .   3    I      71      1,111.31  .         „  ,  . 

V=  -+- 5- tH — ^--r...,  since  P«.(i)  =  i. 

2^4    4      8     2     4'    '   12    2.4    4'        '  *"^  ^ 

To  two  decimal  places  V  =  0.68,  and  the  point  r  =  -,  6  =  0 
is  at  potential  0.68. 

If  r  =  5  and  ^  =  — ,  (18)  and  Table  I,  at  the  end  of  this 
4 

chapter,  give 

^=2^  +  i-T-°-7°7i  +  r~- ^•0-1768+. ..=0.12, 

and  the  point  r  =  5,   0  =  is  at  potential  0.12. 

If  the   radius   of  the  conductor  is  a  instead  of   unity,  we 
have  only  to  replace  r  by  -  in  (17)  and  (18). 


Art.  5.]  PROBLEM  IN  bessel's  functions.  183 

Prob.  2.  One  half  the  surface  of  a  solid  sphere  12  inches  in  di- 
ameter is  kept  at  the  temperature  zero  and  the  other  half  at  100  de- 
grees centigrade  until  there  is  no  longer  any  change  of  temperature 
at  any  point  within  the  sphere.  Required  the  temperature  of  the 
center ;  of  any  point  in  the  diametral  plane  separating  the  hot  and 
cold  hemispheres  ;  of  points  2  inches  from  the  center  and  in  the 
axis  of  symmetry  ;  and  of  points  3  inches  from  the  center  in  a  di- 
ameter inclined  at  an  angle  of  45°  to  the  axis  of  symmetry. 

Ans.  50°;  50°;  73°.9  ;  26°.!  ;  77°.i  ;  22°.9. 

Art.  5.    Problem  in  Bessel's  Functions. 

As  a  last  example  we  shall  take  the  following  problem : 
The  base  and  convex  surface  of  a  cylinder  2  feet  in  diameter 
and  2  feet  high  are  kept  at  the  temperature  zero,  and  the  upper 
base  at  100  degrees  centigrade.  Find  the  temperature  of  a 
point  in  the  axis  one  foot  from  the  base,  and  of  a  point  6  inches 
from  the  axis  and  one  foot  from  the  base,  after  the  permanent 
state  of  temperatures  has  been  set  up. 

If  we  use  cylindrical  coordinates  and  take  the  origin  in  the 
base  we  shall  have  to  solve  equation  (6),  Art.  I  ;  or,  represent- 
ing the  temperature  by  u  and  observing  that  from  the  sym- 
metry of  the  problem  u  is  independent  of  0, 

or         r  or       oz 

(3> 
(4> 

Assume  u  =  RZ  where  R  is  a  function  of  r  only  and  Z  of 
z  only;  substitute  in  (i)  and  reduce. 

...        ^  \  d'R   ^     \   dR  I  d'Z  ,. 

^'^''  R-^r^^VRTr  =  -Z^'  (^> 

The  first  member  of  (5)  does  not  contain  z\  therefore  the 
second  member  cannot.     The  second  member  of  (5)  does  not 


bject  to  the  conditions 

u  ^=  0     when 

z 

=  0, 

u^o 

r 

=--  I, 

Uz=z  IQO      '' 

z 

=  2. 

184  HARMONIC    FUNCTIONS.  [ChaP.  V. 

contain  r;  therefore  the  first  member  cannot.  Hence  each 
member  of  (5)  is  a  constant,  and  we  can  write  (5) 

R  'dr"  ^  rRdr~       'z'dz'  "       ^ '  ^  ^ 

when  ij^  is  entirely  undetermined. 

d'^Z 
Hence  _^-/f'^Z=o,  (7) 

and  -I-  _       o-  ;^'i?  =z  o.  (8) 

dr'        r  dr 

Equation  (7)  is  easily  solved,  and  its  general  solution  is 

Z  —  Aef^'  -{-  Be-^^,  or  the  equivalent  form 

Z  =  C  cosh  {^z)  +  D  sinh  {jaz).  (9) 

We  can  reduce  (8)  slightly  by  letting  ^r  =  x,  and  it  becomes 

d'R   .    \dR   .    ^  '        , 

dx"    ^   xdx^  ^     ^ 

Assume,  as  in  Art.  4,  that  R  can  be  expressed  in  terms  of 
whole  powers  of  x.  Let  R  =  '2a^x''  and  substitute  in  (10). 
We  get 

2[n(n—  i)a„x'' " '  +  na^x''-''  +  ^„;i:"]  =  o, 

an  equation  which  must  be  true,  no  matter  what  the  value  of  x. 
The  coefficient  of  any  given  power  of  x,  as;i;*"^  must,  then, 
vanish,  and 

k{k  —  \)ak  +  M  +  ^^. ,  =  O, 
or  >^"^A  +  ^^_,  =  0, 

whence  we  obtain  Uk..^——  k'Uk  (11) 

as  the  only  relation  that  need  be  satisfied  by  the  coefficients  in 
order  that  R  =  'SUkX''  shall  be  a  solution  of  (10). 

If  k  =.  o,     ^;fe  _ ,  =  o,     ^^  _  ^  =  o,     etc. 

We  can,  then,  begin  with  /^  =  o  as  the  lowest  subscript. 


Art.  5.J  PROBLEM  IN  bessel's  functions.  185 

From  (ii)  ttk—         ^"^ 


Then      «.=  -^,     '^.^^t;^,     ''•=-?:7:6-^'^- 

Hence       7?  =  ..[i  -  |  + -^^  - -^,  +  .  . .  ], 

^here  ^^  may  be  taken  at  pleasure,  is  a  solution  of  (lo),  pro- 
vided the  series  is  convergent. 

-  Take  a^=.  \,  and  then  R  —  JJ^x)  where 

is  a  solution  of  (lo). 

Jlpc)  is  easily  shown  to  be  convergent  for  all  values  real  or 
imaginary  of  x,  it  is  a  new  and  important  form,  and  is  called  a 
Be^^l's  Function  of  the  zero  order,  or  a  Cylindrical  Har- 
Tnonic. 

Equation  (lo)  was  obtained  from  (8)  by  the  substitution  of 
jc  ^n  jAr\  therefore 

is  a  solution    of  (8),   no    matter  what   the   value    of  /^ ;    and 

u  =  JJ^^r)  sinh  {fAz)  and  u  = /^(/ur)  cosh  (/jz)  are  solutions  of 

(i).     u  =  /X^r)  sinh  (^z)  satisfies  condition  (2)  whatever  the 

value  of  //.     In  order  that  it   should    satisfy  condition  (3)  jn 

must  be  so  taken  that 

/o(>")  =  o;  (13) 

that  is,  JJ,  must  be  a  root  of  the  transcendental  equation  (13). 

It  was  shown  by  Fourier  that  /X^)  =  O  has  an  infinite  num- 
ber of  real  positive  roots,  any  one  of  which  can  be  obtained  to 
any  required  degree  of  approximation  without  serious  dififi- 
<:ulty.     Let  //, ,  /z,,  /i^,  .  .  .  be  these  roots  ;  then 

^  =^  A,/o{M,r)  sinh  (//,^)  +  AJl^^r)  sinh  (//,^) 

+  AJlfx/)  sinh  (/i3^)  +  .  .  .    (14) 

is  a  solution  of  (i)  which  satisfies  (2)  and  (3). 


186  HARMONIC    FUNCTIONS.  [ChAP.  V.. 

If  now  we  can  develop  unity  into  a  series  of  the  form 

I  =  BJX^.r)  +  BJiix/)  +  ^3/o(/^3^)  +  .  .  . , 
I 

,  VB,  sinh  {^,z)  j- ,      ^  \  B^  sinh  {u.,2)  ^  ,      .    ,         "1   /   ^\ 

*i  L   sinh  (2/i,)  sinh  (2//,)  J 

satisfies  (i)  and  the  conditions  (2),  (3),  and  (4). 

We  shall  see  later  (Art.  21)  that  lij^x)  =  -  ^^ 

ax 

for  values  of  r  <  i. 
Hence 

u  -  ic,c,[ IA!^^}:^1^A.  4.   /o(M  sinh  {fx^z)  1     /, ^v 

L/.,/X)"0  sinh  (2//,)  "^  /.,/Xa/,)  sinh  (2/.,)  "^  •  •  -J     ^'^/ 

is  our  required  solution. 

At  the  point  r  =  o,  -3:  =  i  (17)  reduces  to 

r  sinh  ^^  .  sinh  /^,  ,         -r 

^  ;^^  200 - H — -r-  I 

l-M^/^(^,)  sinh  (2//J       /^,/,(aO  sinh  (2//J  ^  '  *  *  J 

~         LywyX/^i)  cosh  ;.(^        //,/X/O  cosh  //,       *  *  *J' 

since  y/o)  =  l  and  sinh  {2x)  =  2  sinh  ;r  cosh  x. 

If  we  use  a  table  of  Hyperbolic  functions*  and  Tables  II 
and  III,  at  the  end  of  this  chapter,  the  computation  of  the 
value  of  u  is  easy.     We  have 

/i,  =  2.405  ^,=       5.520 

/liMd  =  0.5 190  /M  =  -  0.3402 


colog     yw,  =  9.6189  —  10     colog     //,  =       9.2581     —10 

"  /X/^O  =  0.2848  -  /.U)=     0.46^3^^ 

**  cosh/i,  =  9.2530  —  10   "  cosh//3=   7-9037  —  10 


9.1567  —  10  7.6301;;  —  10 

*  See  Chapter  IV,  pp.  162,  163,  for  a  four-place  table  on  hyperbolic  func- 


Art.  5.] 


PROBLEM    IN    BESSEL's    FUNCTIONS, 


isr 


{Mx/iM  cosh  /^,)-'  =       0.1434 
(m,/M  cosh  yu,)-^  =  -  0.0058 

0.1376; 

At  the  point  r  =  J,  ^  =  i,  (17),  reduces  to 


L  Mi/r(Mi)  cosh  /^, 


(//J  cosh /^,       /^^/iK)  cosh><3 
/o(i/^.)  -  0.6698 


u  =  if.S 


+ 


^Og/MM^)  =9-8259      -   10 

colog  MiA(Mr)  cosh /J,  =  9.1567    -  10 


8.9826 


10 


^Og/MMa)   =  9-2248//  -    10 

colog  M^/,(Mi)  coshyw,  =       7.6301;?  —  10 


6.8549 


10; 


/oiiM.) 


0.0961 


-V 


Mi/M  cosh  /^, 

/,(iA<J  _  0.0007 . 

Mt/x(Mi)  cosh  //,      0.0968  '         u  =  9°.7 

If  the  radius  of  the  cyHnder  is  a  and  the  altitude  d,  we  have 
only  to  replace  //  by  /xa  in  (13)  ;  2//^,  2//,,  ...  in  the  denomi- 
nators of  (15)  and  (17)  by  M  M  •  •  . ;  and  yw,,  jj,,  ja,,  .  .  .  in 
the  denominators  of  (16)  and  (17)  by  //,<?,  jj^a,  i^i^a,  .... 

Prob.  3.  One  base  and  the  convex  surface  of  a  cylinder  20  cen- 
timeters in  diameter  and  30  centimeters  high  are  kept  at  zero  tem- 
perature and  the  other  base  at  100  degrees  Centigrade.  Find  the 
temperature  of  a  point  in  the  axis  and  20  centimeters  from  the  cold 
base,  and  of  a  point  5  centimeters  from  the  axis  and  20  centimeters 
from  the  cold  base  after  the  temperatures  have  ceased  to  change. 

Ans.   13°- 9;  9°.6. 


188  HARMONIC    FUNCTIONS.  [ChAP,  V. 

Art.  6.    The  Sine  Series. 

As  we  have  seen  in  Art.  3,  it  is  sometimes  important  to  be 
able  to  express  a  given  function  of  a  variable,  x^  in  terms  of  sines 
of  multiples  of  x.  The  problem  in  its  general  form  was  first 
solved  by  Fourier  in  his  *'  Theorie  Analytique  de  la  Chaleur" 
(1822),  and  its  solution  plays  an  important  part  in  most  branches 
of  Mathematical  Physics. 

Let  us  endeavor  to  so  develop  a  given  function  of  x,f{x^, 
in  terms  of  sin  x,  sin  2x,  sin  ^x,  etc.,  that  the  function  and  the 
series  shall  be  equal  for  all  values  of  x  between  o  and  n. 

We  can  of  course  determine  the  coefficients  rf,,  a^,  ci^, .  . .  a^ 
so  that  the  equation 

f(x)  =  a^  sin  x  ■\-  a^  sin  2x  -\-  a^  ?\xi  }^x  -\-  .  .  . -\-  a^  sin  nx   (i) 

shall  hold  good  for  any  n  arbitrarily  chosen  values  of  x  between 
o  and  n\  for  we  have  only  to  substitute  those  values  in  turn 
in  (i)  to  get  71  equations  of  the  first  degree,  in  which  the  n  co- 
efficients are  the  only  unknown  quantities. 

For  instance,  we  can  take  the  n  equidistant  values  Ax,  2 Ax, 

2  Ax,  .  .  .  nAxy  where  Ax  =  — ; — ,  and  substitute  them  for  x  in 

(i).     We  get 

/{Ax)  =    a^  sin  Ax  +    a^  sin  2Ax  -\-    a^  sin  ^^Ax  +  •  • 

+  a^  sin  nAx, 

/[2Ax)  =  a^  sin  2Ax  +    cl^  sin  ^Ax  +    a^  sin  6 Ax  +  .  . 

-|-    ^„  sin  2nAx, 

f(^Ax)  =  «,  sin  2,^^  +    ^a  sin  6Ax  -\-    a^  sin  gAx  +  .  . 

-|-  a^  sin  ytAx, 


K2) 


f{nAx)  =  a^  sin  nAx  +  a^  sin  2nAx  +  ^s  sin  yiAx  +  .  . 

-f-  Un  sin  fi^Ax, 

n  equations  of  the  first  degree,  to  determine  the  n  coefficients 

^,  >  ^a  »  ^3  »  •  •  •  ^«- 

Not  only  can  equations  (2)  be  solved  in  theory,  but  they 
can  be  actually  solved  in  any  given  case  by  a  very  simple  and 


Art.  6  ]  the  sine  series.  189 

ingenious  method  due  to  Lagrange,*  and  any  coefficient  a^  can 
be  expressed  in  the  form 

/{kAx)  sin  {KmAx\  (3) 


n-\-  \^- 

If  now  n  is  indefinitely  increased  the  values  of  x  for  which 
(i)  holds  good  will  come  nearer  and  nearer  to  forming  a  con- 
tinuous set  ;  and  the  limiting  value  approached  by  a„,  will 
probably  be  the  corresponding  coefficient  in  the  series  required 
to  represent  f{x)  for  all  values  of  x  between  zero  and  n. 

Remembering  that  {n  +  \)Ax  —  n,  the  limiting  value  in 
question  is  easily  seen  to  be 


^/^-) 


sin  mxdx,  (4) 


This  value  can  be  obtained  from  equations  (2)  by  the  fol- 
lowing device  without  first  solving  the  equations : 

Let  us  multiply  each  equation  in  (2)  by  the  product  of  Ax 
and  the  coefficient  of  a^  in  the  equation  in  question,  add  the 
equations,  and  find  the  limiting  form  of  the  resulting  equation 
as  n  increases  indefinitely. 

The  coefficient  of  any  a,  a^  in  the  resulting  equation  is 

sin  kAx  sin  mAx .  Ax  +  sin  2k Ax  sin  2m Ax .  Ax  -{-  .  ,  , 
-\-  sin  hkAx  sin  nmAx ,  Ax. 

Its  limiting  value,  since  {n  +  \)Ax  =  -x,  is 


sin  KX  sin  mx .  dx ; 
but 

■K  IT 

C  sin  KX  sin  mx.dx  =  if  [cos  (m  —  k^x  —  cos{m  +  K)x']dx  =  o 
0  0 

if  m  and  k  are  not  equal. 

*  See    Riemann's  Partielle    Differcntialgleichungen,  or   Byerly's    Fourier's 
Series  and  Spherical  Harmonics, 


190  HARMONIC    FUNCTIONS.  [ChAP.  V. 

The  coefficient  of  a^  is 

^;i:(sin''  mAx-\-?AVi  27nAx-\-^\Vi   ^niAx -\-  .  .  .  +sin'  nmAx). 

Its  limiting  value  is 
It 
j  sin"  mx  .dx  =  —, 

0 

The  first  member  is 

f{Ax)  sin  mAx .  Ax  -\-/(2Ax)  sin  2m Ax  .  Ax  + .  .  . 
-\-/{nAx)  sin  mnAx .  Ax, 
and  its  limiting  value  is 

If 
I  f(x)  sin  mx .  dx, 

0 

Hence  the  limiting  form  approached  by  the  final  equation 
as  n  is  increased  is 


/  fix)  sin  mx .  dx  =  —a^ 


Whence  a„  =  -J  A^)  sin  mx.dx  (5) 

as  before. 

This  method  is  practically  the  same  as  multiplying  the 
-equation 

f{x)  =  a^  sin  x  -{- a^  sin  2x  +  ^3  sin  3;i:  +  •  •  •  (6) 

by  sin  mx .  dx  and  integrating  both  members  from  zero  to  n. 

It  is  important  to  realize  that  the  considerations  given  in 
this  article  are  in  no  sense  a  demonstration,  but  merely  estab- 
lish a  probability. 

An  elaborate  investigation  *  into  the  validity  of  the  develop- 
ment, for  which  we  have  not  space,  entirely  confirms  the  results 
formulated  above,  provided  that  between  x  =.0  and  x  ~  n  the 

*  See  Art.  10  for  a  discussion  of  this  question. 


Art.  6  ]  THE    SINE    SERIES.  191 

function  is  finite  and  single-valued,  and  has  not  an  infinite  num- 
ber of  discontinuities  or  of  maxima  or  minima. 

It  is  to  be  noted  that  the  curve  represented  by  y  —  f{x) 
need  not  follow  the  same  mathematical  law  throughout  its 
length,  but  may  be  made  up  of  portions  of  entirely  different 
curves.  For  example,  a  broken  line  or  a  locus  consisting  of 
finite  parts  of  several  different  and  disconnected  straight  lines 
can  be  represented  perfectly  well  by_;^  =  a  sine  series. 

As  an  example  of  the  application  of  formula  (5)  let  us  take 
the  development  of  unity. 

Here  f{x)  =  i. 


cifH^  —  I  sin  mx .  dx ; 


/ 


,  cos  mx 

sm  mx  .dx= . 

m 


f 


sin  mx.dx  =  — (i  —  cos  mn:)  =  — [i  —  (—  l)*"! 
m  ^        m  V        /  J 


=  o  if  /«  is  even 


=  —  \{  m  is  odd. 
m 


TT  4 /sin  X    ,    sin  ^x   ,    sin  ^x    .  s\n  Jx   ,         \     ,  . 

Hence  ,  =  l(^-__  + --^  + —^  +_^  +  .  .  .j.   (7) 

It  is  to  be  noticed  that  (7)  gives  at  once  a  sine  development 
for  any  constant  c.     It  is, 

c  =  l£(^ill£  _L_  ^^"3-y   I    sin  S^   ,         \  /gN 

Prob.  4.  Show  that  for  values  of  x  between  zero  and  tt 

,  .                 Fsin  X      sin  2x  .  sin  -ix      sin  ax    ,          ~] 
(a)     x  =  2 + ^ ^  +  .  .  ■    , 

L.    I  2  3  4  J 

/7\  ^/   \       4  Tsi"  ^      sin  'ix  ,   sin  kx      sin  tx    ,         "1 


192  HARMONIC    FUNCTIONS.  [ChAP.  V., 

\if(x)  =  X  ior  o  <  X  <  -,  and  f{x)  =  n  —  x  ior  -  <  x  <  n, 

{c)  f{x)  = 
2  fsin  ^       2  sin  2^  .  sin  3^  ,   sin  ^x      2  sin  6x      sin  ']x  ~| 

iif{x)  =  1  iox  o  <  X  <  -y  and  f{x)  —  o  (or  -  <  x  <  tt. 


(d)  sinh  X 

= 

2  sinh  TT 
n 

~I      .                  2      . 

-  sin  :r sin 

-2                     5 

(.)  x^  = 

sin  :v sin  2X 

2 

^      '  4 

-^  sin  -ix 

10        ^         17 


+  I -3 jsin  3^ sin 4x-\-.,,    . 


Art.  7.    The  Cosine  Series. 

Let  us  now  try  to  develop  a  given  function  of  ;ir  in  a  series 
of  cosines,  using  the  method  suggested  by  the  last  article. 
Assume 

/{x)  —  b^^b^  cos  X  -\-b^  cos  2x  +  b^  x:os  ^x  -\-  .  .  .  (i). 

To  determine  any  coefficient  b^  multiply  (i)  by  cos  mx .dx 
and  integrate  each  term  from  o  to  n. 

I  b^  cos  mx  .dx=o. 

0 

TT 

/  bk  cos  kx  cos  mx .  dx—o,     if  m  and  /^  are  not  equal. 

0 

TT 

/  b^  cos"  ;/2;ir   /^;ir  =  — ^m,     i^  ^  is  not  zero. 
0 

■n 

Hence  b^  =  —  J A^)  cos  mx.dx,  (2) 

0 
if  ;;/  is  not  zero. 


Art.  7.]  THE   COSINE   SERIES.  193 

To  get  b^  multiply  {\)\iy  dx  and  integrate  from  zero  to  n. 
Jb.dx  —  bjt, 

0 
jr 

/  bk  cos  kx  ,dx  •=.  o. 

0 

It 
Hence  b,  =  - ff{x)dx,  (3) 

0 

which  is  just  half  the  value  that  would  be  given  by  formula  (2) 
if  zero  were  substituted  for  m. 

To  save  a  separate  formula  (l)  is  usually  written 

f{x)  =  ib,  +  b,  cos  x-{-b^  cos  2x  +  b,  cos  3x  +  ,  ,  .,       (4) 

and  then  the  formula  (2)  will  give  b^  as  well  as  the  other  coef- 
ficients. 

Prob.  5.  Show  that  for  values  of  x  between  o  and  tt 

,    ,  7t        A  /cos  X    ,    COS  -IX     ,    COS  KX     ,  \ 

rrx   r/   \       ^      8 /cos  2Ji[:  ,   cos  6:^   ,    cos  10^    ,  \ 

w/w  = -- -(^1-+-^^+-^^^  +  •  •  .j, 

if  /(x)  =  ^  for  o  <  jc  <  -,  and  /(x)  =  tt  —  x  for  —  <  x  X  tt; 

2  2 

,  .     -,  .        1,2  /cos  X      cos  ^x   ,   cos  KX  \ 

w  /M=^+^[-, r  +  5  -  •  •  •]• 

if  /(^)  =  I  for  o  <  jc  <  -,  and  /(x)  =  o  (or  —  <  x  <  tt^ 
2  2 

.21                                I 
(d)  sinh  X  =~    -(cosh  tt  —  i) (cosh  tt  -|-  i)  cos  x 

H — (cosh  Tt  —  i)  cos  2jt: (cosh  TT  +  i)  COS  3J1:  +  •  •  •    ; 

5  10  J 

,  V      a        tt'  /cos  a:       cos  2Jt:       cos  zx       cos  4^    ,  \ 

W"=T-4-^ — r'-+-V — !--  +  •••> 


194  HARMONIC    FUNCTIONS.  [ChAP.  V. 

Art.  8.    Fourier's  Series. 

Since  a  sine  series  is  an  odd  function  of  x  the  development 
of  an  odd  function  of  x  in  such  a  series  must  hold  good  from 
X  —  —  7t  to  X  =  n,  except  perhaps  for  the  value  ;ir  =  o,  where 
it  is  easily  seen  that  the  series  is  necessarily  zero,  no  matter 
what  the  value  of  the  function.  In  like  manner  we  see  that 
if  f{x)  is  an  even  function  of  x  its  development  in  a  cosine 
series  must  be  valid  from  x  =^  —  n  to  x  =^  n. 

Any  function  of  x  can  be  developed  into  a  Trigonometric 
series  to  which  it  is  equal  for  all  values  of  x  between  —  n  and  n, 

Let/(;ir)  be  the  given  function  of  x.  It  can  be  expressed 
as  the  sum  of  an  even  function  of  x  and  an  odd  function  of  x 
by  the  following  device  : 

y^.^)  ^  /(^)  +/(-  ^)  [  -A^)  -A-  ^)  (J) 

2  2 

identically;  but         ^   i^^ 1  jg   not  changed   by  reversing 

the  sign  of  x  and  is  therefore  an  even  function  of  x;  and  when 

we  reverse  the  sign  of  x,  "^-^ "^ is  affected  only  to  the 

extent  of  having  its  sign  reversed,  and  is  consequently  an  odd 
function  of  x. 

Therefore  for  all  values  of  x  between  —  tt  and  tt 

•^^  ^ =  ~0o  +  ^1  COS X  -\-d^  COS  2x  +  ^3  COS  3;r  -f-  .  .  » 

u                   I,         2     ^/{x)+/(-  x) 
where  b^=  —     ^  ^    '  -^  ^ ^  cos  mx  .dx) 

7t  %J  2 

0 

and       -^-^-^ — ^^ =  a,  sm  x-{-a^  sm  2x  -}- a^  sm  ^x -\-  ,  .  . 


,                                                2     f'/{x)  —/{—  x)    . 
where  a^=  -  /  =^^-!- — -^ 1  sin  mx . dx 

7C  J  2 


Art.  8.]  Fourier's  series.  195 

^,„  and  a^  can  be  simplified  a  little. 

o^~  —  /  -"^  >   '  -"^ fLcos  mx ,  dx 

71  d  2, 

0 

n  IT 

=  —     //W  COS  mx  .  dx-\- 1  f{—x)  cos  mx  .dx   ; 

0  0  -J 

but  if  we  replace  ;«;  by  —  ;r,  we  get 

IT  — IT  0 

/  /(—  x)  cos  mx .  dx=—J  fix)  cos  mx,dx=  I  f{x)cos  mx,dx, 

n- 

and  we  have  bm  =  —  I  f{^)  cos  mx .  ^.r. 

—  IT 

In  the  same  way  we  can  reduce  the  value  of  ^^  to 
—  /  f{x^  sin  mx .  dx, 

—IT 

Hence 

f{x)  =-  b^-\-  b^  cos  X  -\-  b^  cos  2;tr  +  ^3  cos  ^x  -{-, ,, 

+  ^,  sin  X  -\-  a^  sin  2;r  +  ^3  sin  S^r  +  •  •  •  >    (2) 

IT 

where  ^m  =  —  I  fi.^)  cos  mx  .  ^;i:,  (3) 


and 


^«  =  —  /  y(;r)  sin  mx .  ^^r,  (4) 


and  this  development  holds  for  all  values  of  x  between  —  7t 

and  n. 

The  second  member  of  (2)  is  known  as  a  Fourier's  Series. 

The  developments  of  Arts.  6  and  7  are  special  cases  of 
development  in  Fourier's  Series. 

Prob  6.  Show  that  for  all  values  of  x  from  —  ;r  to  tt 

2  sinh  nr-i       i                 i                    i                     i  H 

^= cos  ^  H — cos  2^ cos  3:«:H cos4J<:+..o 

7t  \_2  2  5  10  "^  17 


196  HARMONIC    FUNCTIONS.  [ChAP.  V. 

2  sinh  Tt  Fi    .             2.           ,3-               4-  il 

-\ -sin^ sin  2^  +  —  sin  3^: sm  4;^  +  . . .    . 

^  Tt  \_2  5  10  17  J 

Prob.  7.  Show  that  formula  (2),  Art.  8,  can  be  written 

A^)  =  1^0  COS^„  +  C,  cos  (X  -  A)  +  ^.  cos  (2X  -  /3^) 

+  c,  cos  (3^  -  A)  +  . .  • , 
where  c^,  =  {aj  +  bj)^     and     ^^  =  tan'^  -—' 

Prob..  8.  Show  that  formula  (2),  Art.  8,  can  be  written 

f{x)  =-c,  sin  P,  +  c,  sin  {x  +  ^,)  -f  ^,  sin  {2X  +  ^,) 
2 

+  ^,  sin  (3:^4- /?3)  +  .  .., 


where  <;„,  =  {a  J  +  z^^")*     and     A«  =  tan 


a. 


Art.  9.    Extension  of  Fourier's  Series. 

In  developing  a  function  of  x  into  a  Trigonometric  Series  it 
is  often  inconvenient  to  be  held  within  the  narrow  boundaries 
X  =^  —  7t  and  X  =z  7t.     Let  us  see  if  we  cannot  widen  them. 

Let  it  be  required  to  develop  a  function  of  x  into  a 
Trigonometric  Series  which  shall  be  equal  to  f{x)  for  all  values 
of  X  between  x  ^^  —  c  and  x  =^  c. 

Introduce  a  new  variable 

n 

z  =  — X, 
c 

which  is  equal  to  —  tt  when  x  ^=z  —  c^  and  to  n  when  x  =^  c. 

f{x)  =  f\—^ )  ^^^  ^^  developed  in  terms  of  z  by  Art.  8, 
(2),  (3),  and  (4).     We  have 

/(^^)  =  -  ^0  -h  <^i  cos  ^  +  b^  cos  2z  +  z^,  cos  3^  +  , . , 

-\-  a^  sin  z  -\-  a^  sin  2^  -|-  ^3  sin  3^  +  . . .,     (i) 
If 
where  b^=^  —  / / ( — z\  cos  mz .  dz,  (2) 


Art.  9.]  EXTENSION  OF  Fourier's  series.  197 

and  a„,  =  -~J*/[-~^)  sin  ms .  d^,  --         (3) 

—n 

and  (i)  holds  good  from  2  =^  —  7t  to  ^  =  tt. 

Replace  2  by  its  value  in  terms  of  x  and  (1)  becomes 

/W   =    2    ^0  +  ^'  ^^^  — +  ^.  cos— ^+^3COS-y-+... 

+  ^,  sin  -^+ih  sin  -7-  +  «3  sin  -y  +•  •  •  I    (4) 
and  (2)  and  (3)  can  be  transformed  into 

bm  =  —J  /W  cos  —^dx,  (5) 

—c 

a^  =  —  f/(x)  sin  ^^^dx,  (6) 

C  t/  c 

—c 

and  (4)  holds  good  from  ;i:  =  —  c  to  x  ^=  c. 

In  the  formulas  just  obtained  c  may  have  as  great  a  value 
as  we  please  so  that  we  can  obtain  a  Trigonometric  Series  for 
f(x)  that  will  be  equal  to  the  given  function  through  as  great 
an  interval  as  we  may  choose  to  take. 

It  can  be  shown  that  if  this  interval  c  is  increased  indefi- 
nitely the  series  will   approach  as  its  limiting  form  the  double 

00  00 

integral  —  /  f{\)dk  /  cos  a{X  —  x)da^  which   is   known   as  a 

—  CO  0 

Fourier's  Integral.     So  that 

f{x)  =  -^y'/('^)^'^Aos  a(X  -  x)da  (7) 

—00  0 

for  all  values  of  x. 

For  the  treatment  of  Fourier's  Integral  and  for  examples 
of  its  use  in  Mathematical  Physics  the  student  is  referred  to 
Riemann's  Partielle  Differentialgleichungen,  to  Schlomilch's 
Hohere  Analysis,  and  to  Byerly's  Fourier's  Series  and 
Spherical  Harmonics. 


198  HARMONIC    FUNCTIONS.  [ChAP.  V. 

Prob.  9.  Show  that  formula  (4),  Art.  9,  can  be  written 
f{x)  =  ^-c,  cos  ft,  +  c,  cos  (y^  -  A  )  +  ^«  cos  {^-^  -  ^,] 

+  .3  cos  (^^--^3)+..., 

where  €„,  =  {aj  +  bm^     and     ^m  =  tan"^  —, 

Prob.  10.  Show  that  formula  (4),  Art.  9,  can  be  written 
f{x)  =  i.,  sin  /J.  +  ^,  sin  (^  +  ^,)  +  .,  sin  {—  +  /S,) 

+  .,  sin  (i^ +  /?.)+..., 

where  <:„=(»„'  +  *„')*     and    /?„  =  tan-'-^. 


Art.  10.    Dirichlet's  Conditions. 

In  determining  the  coefficients  of  the  Fourier's  Series  rep- 
resenting f{x)  we  have  virtually  assumed,  first,  that  a  series  of 
the  required  form  and  equal  to  fix)  exists ;  and  second,  that 
it  is  uniformly  convergent ;  and  consequently  we  must  regard 
the  results  obtained  as  only  provisionally  established. 

It  is,  however,  possible  to  prove  rigorously  that  if /(^)  is 
finite  and  single-valued  from  x^=^— n\.Qx^=Tt  and  has  not 
an  infinite  number  of  (finite)  discontinuities,  or  of  maxima  01 
minima  between  x  ^=  ~  n  and  ;r  =  tt,  the  Fourier's  Series  of 
(2),  Art.  8,  and  that  Fourier's  Series  only,  is  equal  to  f(x) 
for  all  values  of  x  between  —  n  and  tt,  excepting  the  values  of 
X  corresponding  to  the  discontinuities  of  y(;r),  and  the  values 
n  and  —  n ;  and  that  if  ^  is  a  value  of  x  corresponding  to  a 
discontinuity  of  f{x\  the  value  of  the  series  when  ;tr  =r  ^  is 

—  g  _  Q  [/(^  +  e)  -J-  f{c  —  e)  ]  ;    and    that   when    x  =  n    ox 
X  =  —7t  the  value  of  the  series  is  |-[/(^)  +/(—  ^)]. 

This  proof  was  first  given  by  Dirichlet  in  1829,  and  may  be 
found  in  readable  form  in  Riemann's  Partielle  Differential- 
gleichungen  and  in  Picard's  Traits  d'Analyse,  Vol.  I. 


DIRICHLET  S    CONDITIONS. 


199 


Art.  10.] 

A  good  deal  of  light  is  thrown  on  the  peculiarities  of  trigo- 
nometric series  by  the  attempt  to  construct  approximately  the 
curves  corresponding  to  them. 

If  we  construct  7  =  ^,  sin  x  and  y  ^^  a^  sin  2x  and  add  the 
ordinates  of  the  points  having  the  same  abscissas,  we  shall  ob- 
tain points  on  the  curve 

y  =  a^  s\x\  X  -\-  a^  sin  2x. 

If  now  we  construct  J/  =  a^  sin  t^x  and  add  the  ordinates  to 
those  of  j^  =  «,  sin  x  -\-  a^  sin  2x  we  shall  get  the  curve 

^  =  ^j  sin  ;i;  4-  a^  sin  2x  -f  a^  sin  3;r. 

By  continuing  this  process  we  get  successive  approximations  to 

y  =  a^  sin  X  -\-  a^  sin  2x  -{-  a^  sin  ^x  -\-  a^sAVi  /i^x  -\-  •  •  • 


Y 

0 

/       \ 

X 

/ 

Tt 

I 

\ 

Y 

0 

lA  -  .>\ 

X 

i 

II 

^ 

Y 

0 

/;  :: 

"^^5^ 

X 

1 

III 

Tt 

Let  us  apply  this  method  to  the  series 
jj/  =  sin  ;t:  +  I  sin  3^  +  I  sin  5;ir  +  . .  . .  (i)  (See  (7),  Art.  6.) 

y  =  o  when  x  z=  o^  ~  from  x  =  oto  x  =  n,  and  o  when  x  =  n, 
4 

It  must  be  borne  in  mind  that  our  curve  is  periodic,  hav- 
ing the  period  27r,  and  is  symmetrical  with  respect  to  the 
origin. 

The  preceding  figures  represent  the  first  four  approxima- 


200  HARMONIC    FUNCTIONS.  [Chap.  V. 

tion  to  this  curve.  In  each  figure  the  curve  y  =  the  series, 
and  the  approximations  in  question  are  drawn  in  continuous 
hues,  and  the  preceding  approximation  and  the  curve  corre- 
sponding to  the  term  to  be  added  are  drawn  in  dotted  lines. 

Prob.  II.  Construct  successive  approximations  to  the  series 
given  in  the  examples  at  the  end  of  Art.  6. 

Prob.   12.  Construct  successive  approximations  to  the  Maclaurin's 

3  5 

Series  for  sinh  x,  namely  x  -{ ~  H j  -\ 

Art.  11.    Applications  OF  Trigonometric  Series. 

(a)  Three  edges  of  a  rectangular  plate  of  tinfoil  are  kept 
at  potential  zero,  and  the  fourth  at  potential  I.  At  what  po- 
tential is  any  point  in  the  plate? 

Here  we  have  to  solve  Laplace's  Equation  (3),  Art.  i, 
which,  since  the  problem  is  two-dimensional,  reduces  to 


d^'^    d^? 


=  o,  (I) 


subject  to  the  conditions  V  =  o  when  x  =  o,  (2) 

V=o     ''       X  =  a,  (3) 

F  =  o     "      J/  =  o,  (4) 

F  =  I     "      J  =  ^.  (5) 

Working  as  in  Art.  3,  we  readily  get  sinh/?^  sin /3x, 
sinh  /3j/  cos  /3x,  cosh  ^y  sin  /3x,  and  cosh  /3y  cos  /3x  as  particu- 
lar values  of  F  satisfying  (i). 


sinh  —^  sin  satisfies  (i),  (2),  (3),  and  (4). 

Try  ,       ^Tty 


^  psinh  ^                       sinh-^  n 

i      ^sin^+^ ^.in?ZL-+... 


a  a 


is  the  required  solution,  for  it  reduces  to  I  when  y  =z  b.     See 
(7),  Art.  6. 


y  =  o 

when 

X  =  0, 

y  =  o 

<( 

x  =  l, 

^^ 

t  =  0, 

y  =fx    " 

/    =  0. 

Art.  11.]  APPLICATIONS    OF    TRIGONOMETRIC    SERIES.  201 

(5)  A  harp-string  is  initially  distorted  into  a  given  plane 
curve  and  then  released  ;  find  its  motion. .^ 

The  differential  equation  for  the  small  transverse  vibrations 
of  a  stretched  elastic  string  is 

df  ~  "  dx' '  ^'^ 

as  stated  in  Art.  i.     Our  conditions  if  we  take  one  end  of 
the  string  as  origin  are 

(2) 

(3) 

(4) 

(5) 

Using  the  method  of  Art.  3,  we  easily  get  as  particular  solutions 
of  (I) 

y  =  sin  ^x  sin  a/3/,  y  =  sin  /3x  cos  aj3t, 

y  =  cos  fix  sin  a  fit,     and    y  =  cos  fix  cos  a  fit. 

.    mnx        mnat 
y  =  sm  —J-  cos— ^  satisfies  (i),  (2),  (3),  and  (4). 

IJir         .    mnx        mnat  /^rx 

y  ^^ant  sm  — ^  cos  — ^,  \P) 


where  ^'«  =  7  /'/W  sin  ^^ .  dx  (7) 

0 

is  our  required  solution  ;  for  it  reduces  to  f(x)  when  ^  =  o.    See 
Art.  9. 

Prob.  13.  Three  edges  of  a  square  sheet  of  tinfoil  are  kept  at 
potential  zero,  and  the  fourth  at  potential  unity  ;  at  what  potential 
is  the  centre  of  the  sheet  ?     Ans.  0.25. 

Prob.  14.  Two  opposite  edges  of  a  square  sheet  of  tinfoil  are 
kept  at  potential  zero,  and  the  other  two  at  potential  unity  ;  at 
what  potential  is  the  centre  of  the  sheet  ?     Ans.  0.5. 

Prob.  15.  Two  adjacent  edges  of  a  square  sheet  of  tinfoil  are 


202  HARMONIC    FUNCTIONS.  [Chap.  V. 

kept  at  potential  zero,  and  the  other  two  at  potential  unity.  At 
what  potential  is  the  centre  of  the  sheet  ?     Ans.  0.5. 

Prob.  16.  Show  that  if  a  point  whose  distance  from  the  end  of  a 

harp-string  is  -th  the  length  of   the  string  is  drawn  aside  by  the 

player's  finger  to  a  distance  b  from  its  position  of  equilibrium  and 
then  released,  the  form  of  the  vibrating  string  at  any  instant  is  given 
by  the  equation 

2bfi^       !!^  f  I     .    trnt    .    mitx         innat\ 


in-  i)7t'^ 

\  '  m=l 


/  I      .    m7t    .    mTtx         innat\ 
— ^  sm  —  sm  — ~  cos  — -—  , 


Show  from  this  that  all  the  harmonics  of  the  fundamental  note  of 
the  string  which  correspond  to  forms  of  vibration  having  nodes  at 
the  point  drawn  aside  by  the  finger  will  be  wanting  in  the  complex 
note  actually  sounded. 

Prob.  17.*  An  iron  slab  10  centimeters  thick  is  placed  between  and 
in  contact  with  two  other  iron  slabs  each  10  centimeters  thick.  The 
temperature  of  the  middle  slab  is  at  first  100  degrees  Centigrade 
throughout,  and  of  the  outside  slabs  zero  throughout.  The  outer 
faces  of  the  outside  slabs  are  kept  at  the  temperature  zero.  Re- 
quired the  temperature  of  a  point  in  the  middle  of  the  middle  slab 
fifteen  minutes  after  the  slabs  have  been  placed  in  contact. 
Given  a^  =  0.185  i^  C.G.S.  units.     Ans.    io".3. 

Prob.  18.*  Two  iron  slabs  each  20  centimeters  thick,  one  of  which 
is  at  the  temperature  zero  and  the  other  at  100  degrees  Centigrade 
throughout,  are  placed  together  face  to  face,  and  their  outer  faces 
are  kept  at  the  temperature  zero.  Find  the  temperature  of  a  point 
in  their  common  face  and  of  points  10  centimeters  from  the  com- 
mon face  fifteen  minutes  after  the  slabs  have  been  put  together. 

Ans.  22°.8  ;  15".!  ;  i7°.2. 

Art.  12.t    Properties  of  Zonal  Harmonics. 

In  Art.  4,  -3"  =  Pm{^)  was  obtained  as  a  particular  solution  of 
Legendre's  Equation  [(7),  Art.  4]  by  the  device  of  assuming 
that  2  could  be  expressed  as  a  suni  or  a  series  of  terms  of 
the   form   a^x*"  and   then    determining   the    coefificients.     We 

*  See  Art.  3. 

f  The  student  should  review  Art.  4  before  beginning  this  article. 


Art.  12.]  PROPERTIES    OF    ZONAL    HARMONICS.  20S 

can,  however,  obtain  a  particular  solution  of  Legendre's  equa- 
tion by  an  entirely  different  method. 

The  potential  function  for  any  point  {x,  y,  z)  due  to  a  unit 
of  mass  concentrated  at  a  given  point  (^i,7i,  ^,)  is 

V=    ,  ^ (i) 

and  this  must  be  a  particular  solution  of  Laplace's  Equation 
[(3),  Art.  i],  as  is  easily  verified  by  direct  substitution. 
If  we  transform  (i)  to  spherical  coordinates  we  get 

V=    ,  '  -  (2) 

Vr^  —  2rr,[cos  d  cos  0^  -\-  sin  6  sin  6^  cos  (0—0,)]  +  r/ 

as  a  solution  of  Laplace's  Equation  in  Spherical  Coordinates 
[(5),  Art.  I]. 

If  the  given  point  {x^,  /,,  ^,)  is  taken  on  the  axis  of  X,  as  it 
must  be  in  order  that  (2)  may  be  independent  of  0,  0^  =  o,  and 

u) 


4/r'  —  2rr,  cos  0  +  r,' 

is  a  solution  of  equation  (i),  Art.  4. 
Equation  (3)  can  be  written 


d  if  r  is  less  than  r,  ( i  —  2—  cos  <9  +  —  j      can  be  developed 


into  a  convergent  power  series.     Let  ^p^—^  be  this  series, 

I  r^ 

p„,  beinpr  of  course  a  function  of  Q,     Then  V  = —^p,^ —  is  a 

'    \  1 

solution  of  (i).  Art.  4. 

Substituting  this  value  of  V  in  the  equation,  and  remem- 
bering that  the  result  must  be  identically  true,  we  get  after  a 
slight  reduction 


204  HARMONIC    FUNCTIONS.  [ChAP.  V. 

but,  as  we  have  seen,  the  substitution  oi  x  =  cos  6  reduces  this 
to  Legendre's  equation  [(7),  Art.  4].  Hence  we  infer  that  the 
coefficient  of  the  mth  power  of  ^  in  the  development  of 
(i  —  ixz-^z"")-^  \z  a  function  of  x  that  will  satisfy  Legendre's 
equation. 

(l   -  2XZ  +  zy-i  =  [l  -  2{2X  -  Z)]  -*, 

and  can  be  developed  by  the  Binomial  Theorem  ;  the  coefficient 
of  z"^  is  easily  picked  out,  and  proves  to  be  precisely  the  func- 
tion of  X  which  in  Art.  4  we  have  represented  by  P„J^^),  and 
have  called  a  Surface  Zonal  Harmonic. 
We  have,  then, 

<i-2^^+^')-*=^.W+^.W-s+-P.W-^+^.W.*'+ ...  (5) 

if  the  absolute  value  of  z  is  less  than  i. 
l(  X  =  I,  (5)  reduces  to 

(,_2z+zYi  =  P,(i)+P,{i).z  +  PM^^'+PM.^'+..^; 

but     (I  -2z  +  z')-i  =  {i  -z)-'=i+z  +  z'  +  z'  +  ..,; 

hence  ^«X0  =  i-  (6) 

Any  Surface  Zonal  Harmonic  may  be  obtained  from  the 
two  of  next  lower  orders  by  the  aid  of  the  formula 

{n+i)P„  +  ,{x)-{2n+i)xP^{x)  +  nP„_,(x)=o,        (7) 

which  is  easily  obtained,  and  is  convenient  when  the  numerical 
value  of  X  is  given. 

Differentiate  (5)  with  respect  to  z,  and  we  get 

^^^^=^^^  =  P,W  +  2P,(.)..  +  3P.W.. '  +  .... 
whence 

or  by  (5) 

(I  -  2XB  +  z')(PXx)  +  2P,{x)  .  z  +  ^Plx)  .z'..-) 

+  («  —  ^)(^.(^)  +  ^.W  •  ^  +  ^,W  .  2'  +•••■)  =  O.     (8) 


Art.  13.J  problExMS  in  zonal  harmonics.  205' 

Now  (8)  is  identically  true,  hence  the  coefficient  of  each 
power  of  z  must  vanish.  Picking  out  the  coefficient  of  z"^  and 
writing  it  equal  to  zero,  we  have  formula  (7)  above. 

By  the  aid  of  (7)  a  table  of  Zonal  Harmonics  is  easily  com- 
puted since  we  have  PJ^x)  =  I,  and  P^i^)  =  x.  Such  a  table 
for  X  =  cos  6  is  given  at  the  end  of  this  chapter. 

Art.  13.    Problems  in  Zonal  Harmonics. 

In  any  problem  on  Potential  if  Fis  independent  of  0  so 
that  we  can  use  the  form  of  Laplace's  Equation  employed  in' 
Art.  4,  and  if  the  value  of  Fon  the  axis  of  X  is  known,  and 

can  be  expressed  as  2a^r*^  or  as  '^  ~^>  ^'^  can  write  out 

our  required  solution  as 

V=2a„r''P„(cose)     or     V  =  '^'-^,^; 

for  since  P„»(i)  =  i  each  of  these  forms  reduces  to  the  proper 
value  on  the  axis ;  and  as  we  have  seen  in  Art.  4  each  of  them 
satisfies  the  reduced  form  of  Laplace's  Equation. 

As  an  example,  let  us  suppose  a  statical  charge  of  M  units 
of  electricity  placed  on  a  conductor  in  the  form  of  a  thin  circu- 
lar disk,  and  let  it  be  required  to  find  the  value  of  the  Poten- 
tial Function  at  any  point  in  the  "  field  of  force  "  due  to  the 
charge. 

The  surface  density  at  a  point  of  the  plate  at  a  distance  r 

from  its  centre  is 

M 


a  = 


^an  Vd^ 


and  all  points  of  the  conductor  are  at  potential .   See  Pierce's 

Newtonian  Potential  Function  (§  61). 

The  value  of  the  potential  function  at  a  point  in  the  axis 
ot  the  plate  at  the  distance  x  from  the  plate  can  be  obtained 
without  difficulty  by  a  simple  integration,  and  proves  to  be 


M  ,x 


2a  X  -\-  a 


206  HARMONIC    FUNCTIONS.  [Chap.  V. 

The  second  member  of  (i)  is  easily  developed  into  a  power 
series. 

cos  -  ^  — — r 

2a  x^  -\-  a 

MVn        X    .     x'         x'         x'  1  -r       ^       /  \ 

=  — -3  —  3— .  +  Z-?  —  .  .  .    \  i(  X  <  a  (2) 

a  V.2         a    '    3^'        S^         7^  -1  ^       v  ^^ 

MVa  a'      .     a"  a'     .  "I  -r      ^  ,  x 

=  TLJ--^^  +  ^--^^+   ...Jif^>..       (3) 


Hence 


^-1/',(cose)  +  ^^P.(cosS) 


«  L2        a      ^  I  a 

-  iJ>.(cos6>)+...]  (4) 

is  our  required  solution  i(  r  <,a  and  6^  <  -,  as  is 


a 


"^-   ^^P,(cos^)  +  i-5p,(cos^) 
-r  I    r^  5  ^ 


i5p.(cos^)  +  ...]ifr>^.    (5) 
7  r  J 


The  series  in  (4)  and  (5)  are  convergent,  since  they  may  be 
obtained  from  the  convergent  series  (2)  and  (3)  by  multiplying 
the  terms  by  a  set  of  quantities  no  one  of  which  exceeds  one 
in  absolute  value.  For  it  will  be  shown  in  the  next  article  that 
P^  (cos  Q)  always  lies  between  i  and  —  i. 

Prob.  19.  Find  the  value  of  the  Potential  Function  due  to  the 
attraction  of  a  material  circular  ring  of  small  cross-section. 

The  value  on  the  axis  of  the  ring  can  be  obtained  by  a  simple 

integration,  and  is     ,  ,         .  if  M  is  the  mass  and  c  the  radius  of  the 
ring.     At  any  point  in  space,  if  r  <  ^ 

V  =  y  [a(cOS  ti)-\  ^V,(cOS  ^)  +  ^  ^A(C0S  V)  -  . .  .], 

and  if  r  >  ^ 


Art.  14.]  additional  forms.  207 

F=  —  f-^oCcos  6)  -  -  -,P.(cos  6)  +1^  i^FX^os  ^)  -  . .  .1. 
c   V_r  2  r      -  2  .  4r-     *'  J 

Art.  14    Additional  Forms. 

{a)  We  have  seen  in  Art.  I2  that  P„i{x)  is  the  coefficient  of 
z^  in  the  development  of  (i  —  2x2  +  2^)-^  in  a  power  series. 

(1  _  2x2  +  ^^)  -  i  =  [I  -  2{e^^  -\-  e-  «0  +  2'^^ 
=  (i  —  2e^^)-  \i  —  2e-^')  -K 

If  we  develop  (i  —  2e^')-^  and  (i  —  2e-^')-i  by  the  Bi- 
nomial Theorem  their  product  will  give  a  development  for 
(i  —  2x2  -\-  2'')-^.  The  coefficient  of  2"*"  is  easily  picked  out 
and  reduced,  and  we  get 

P.n{cos  6)  = 

I .  "^ .  5  .  .  .  (2m  —  i)r  ^  .  I  ,m  , 

^    ^    .    ^ 2  cos  md  +  2  — -. r  cos  {m  —  2)6 

2 .4.6 ,  .  .  2m        L  I .  {2m  —  i)         ^  ^ 

.    3-M>«--)       cos(.»-4)g  +  ...1  (I) 

~    I  .2.(2w— 1)(2;;^  —  3)        ^  t/      I         J  \j 

If  ;;/  is  odd  the  parenthesis  in(i)  ends  with  the  term  con- 
taining cos  0  ;  if /;^  is  even,  with  the  term  containing  cos  o,  but 
in  the  latter  case  the  term  in  question  will  not  be  multiplied  by 
the  factor  2,  which  is  common  to  all  the  other  terms. 

Since  all  the  coefficients  in  the  second  member  of  (i)  are 
positive,  Pm{cos  0)  has  its  maximum  value  when  6  =  0,  and  its 
value  then  has  already  been  shown  in  Art.  12  to  be  unity. 
Obviously,  then,  its  minimum  value  cannot  be  less  than  —  i. 

(^)  If  we  integrate  the  value  of  P„^{x)  given  in  (11),  Art.  4, 

m  times  in  succession  with  respect  to  x,  the  result  will  be 

J    ■2    c        {21ft I^ 

found  to  differ  from     '  "^ '  ^  '/       . ^(^'  —  ^Y  by  terms  in- 

{2m)\  ^ 

volving  lower  powers  of  x  than  the  ;«th. 

Hence  />„(.)  =  ^_L,  ^^J.' -  ,)».  (2) 


208  HARMONIC    FUNCTIONS.  [ChAP.  V. 

{c)  Other  forms  for  P^{^),  which  we  give  without  demon- 
stration, are 

p  (^)  _  tur  dl ■  (3) 

P^{^)  =  '-fl^+*'^^^-<'°^<PTd^-  (4) 

0 

^».W  -  ^  /p ^7^^ T^.-  (I) 

^e/    [;ir  —  y  ;ir'  —  I  .  COS  0]*"+* 

(4)  and  (5)  can  be  verified  without  difficulty  by  expanding 
and  integrating. 


Art.  15.    Development  in  Terms  of  Zonal  Harmonics. 

Whenever,  as  in  Art.  4,  we  have  the  value  of  the  Potential 
Function  given  on  the  surface  of  a  sphere,  and  this  value  de- 
pends only  on  the  distance  from  the  extremity  of  a  diameter, 
it  becomes  necessary  to  develop  a  function  of  6  into  a  series 
of  the  form 

^„P„(cos  6)  +  A.PXcos  6)  +  A^Plcos  6^)  +  .  .  . ; 

or,  what  amounts  to  the  same  thing,  to  develop  a  function  of 
X  into  a  series  of  the  form 

A^P^x)  +  A.Pix)  +  A^Plx)  +  .  .  . . 

The  problem  is  entirely  analogous  to  that  of  development 
in  sine-series  treated  at  length  in  Art.  6,  and  may  be  solved  by 
the  same  method. 

Assume   f{x)  =  A,P,{x)  +  A,P,{x)  +  A.P^x)  + .  .  .  (i) 

for  —  I  <  ;ir  <  I.     Multiply  (i)  by  PJ^x)dx  and  integrate  from 
--  I  to  I.     We  get 

1  _  1 

ff(x)PS.x)dx  ="£lA„  f  PJ,x)Plx)dx\  (2) 


Art.  16.]  FORMULAS    FOR    DEVELOPMENT.  209" 

We  shall  show  in  the  next  article  that 

1 
j  PJ^x)P„{x)dx  =  o,     unless  m  =  n, 
-1 

and  that     /[^»W]V4.- =  -;^. 

Hence  A„  ==  ^-^^^  f f{x)PJ^x)dx.  (3). 

-1 

It  is  important  to  notice  here,  as  in  Art.  6,  that  the  method 
we  have  used  in  obtaining  A^  amounts  essentially  to  deter- 
mining A^^  so  that  the  equation 

Ax)  =  A,Plx)  +  A,P,l,x)  +  A^Plx)  +  .  .  .  +  A„P„{x) 

shall  hold  good  for  n  -\-  i  equidistant  values  of  x  between  —  I 
and  I,  and  taking  its  Hmiting  value  as  n  is  indefinitely  in- 
creased. 

Art.  16.    Formulas  for  Development. 
We  have  seen  in  Art.  4  that  z  =  Pm{^)  is  a  solution   of 

d  r  d'^z~\ 

Legendre's  Equation  ^    (i  —  x^)  -~  \-\-m(m  +  i)'3^  =  o.    (i> 

dPJx)~] 
( ^  -  ^')  -^  J  +  ^(^  +  I  )^^«  W  =  o,       (2) 

(^-^')-J^~]+H^  +  i)^n{^)    =0.       (3) 


Hence        -7- 
dx 


and  -7- 

dx 


Multiply  (2)  by  PJ^x)  and  (3)  by  P^{x),  subtract,  transpose, 
and  integrate.     We  have 

1 
\m{nf\-  l)-n{n+  l)-\fp„{x)P,(x)dx 


'h<b-^''^']-' 


210 


HARMONIC    FUNCTIONS. 


_„dP^x) 
dx 


[Chap.  V. 
dx     (4) 


=  [^-W(^  -  --y-^"-  -  ^''('^)(^  -  -^)      dx 


-1      ^ 

»/  dx         dx 


.  ^r 


by  integration  by  parts, 


=  o. 


Hence  J PJ^x)P^{x)dx  =  O, 


(6) 


unless  m  =^  n. 

If  in  (4)  we  integrate  from  ;tr  to  I  instead  of  from  —  i  to  i, 
we  get  an  important  formula. 


J  P^(x)P,(x)d^ 


(■->K>^4;-'-''-<'>^] 


m{m-\-  i)  —  n{n-\-  i) 
and  as  a  special  case,  since  PX^)  =  i- 

^     dx 


,  (7) 


/PJx)dx  = 1 r , 
^                     m(m  +  i) 


(8) 


unless  m  =  o. 
1 


To  get  f[P^{x)Ydx  is  not  particularly  difficult.      By  (2), 


Art.  14, 


/[/'^Wiv.-^w^/-^ 


d'^ix'  -  i)^    d^{x'  -  ly 


dx' 


dx'"" 


.  dx  (9). 


By  successive  integrations  by  parts,  noting  that 


d^- 


-^— -(^r'^  —  i)*"  contains  {pc^  —  i)"  as  a   factor  if   /c  <  in,  and 


Art.  16.]  FORMULAS   FOR   DEVELOPMENT.  211 

that  — ^——^ —  =  (2ni)\  we  get  ^  __  _ 


dx 


fip.{^)Yd.  ^  ^-^Qp^  y V  -  i)v^.       (10) 

-1  ^    '     -1 

1  1 

J\j^-  _  i^dx  =  J(x  -  iY{x  +  i)'^dx 
-1  -1 

1 

=  — ^—  r{x  -  i)^-\x + lY'^-'dx 


1 

1 


Hence  /[/^^(.)]V.  = --1_.  („) 

-1  ' 

Prob.  2o.  Show  that  /  F„^{x)dx  =  o  if  /«  is  even  and  is  not  zero 


0 


m—\ 


=  (-  i)^  -T-'n— X  •      ^•^•^T'^    ,  if  ^  is  Odd. 
m{m-\-i)    2  .4.  4  .. .  (w  — i) 

1 

y»                                  I 
[/'^(^)]V:x:  =  , — .     Note  that 

0 
[PtH{x)Y  is  an  even  function  of  x. 

Prob.  22.  Show  that  if   f{x)  =  o  from  :v  =  —  i  to  jc  =  o,  and 
y(^)  =  I   from  jc  =  o  to  x  =■  1, 

/(.)  =  i+J/'.w-Z-.ii'.(.)  +  ii.Lj^.(.)_... 

m=oo 

Prob.  23.  Show  that  F(6)  =  :E  B^F ^{cos  6)     where 

»r 

^^  =  ,3!1±1  fF(6)F Jcos  0)  sin  6  dS. 


2 

0 


212  HARMONIC    FUNCTIONS.  [Chap.  V. 

Prob.  24.  Show  that 

CSC   ^  ::=    ^[l   +  5(^)'^,  (cos  6)  +  9(^-'^)'a(cOS   ^)  +  .  .  .]. 

See  (1),  Art.  14. 

Prob.  25.   Show  that 

2.4  _J 

1  1 

x''Pn{x)dx  = /  x""  — 5^—— — ^^.^,  and  use  the 

-1  -1 

method  of  integration  by  parts  freely. 

Prob.  26.  Show  that  if  F  is  the  value  of  the  Potential  Function 
at  any  point  in  a  field  of  force,  not  imbedded  in  attracting  or  repel- 
ling matter;  and  if  V  —  f{.9)  when  r  =  a, 

V  =  2A,~  FJcos  e)[i  r<a 
and  V  =  ^A^-^;^^P^{cos  0)ii  r>  a^ 

n 

where  A^  =  ^''' J"  '  /7(^)-P^(cos  0)  sin  Odd.  r: 


Prob.  27.  Show  that  if 


V  =  c  when  r  —  a\     F  =  c  if  r  <  a,    and  F=  —  \{  ry-  a. 

r 


Art.  17.     Formulas  in  Zonal  Harmonics. 

The  following  formulas  which  we  give  without  demonstra- 
tion may  be  found  useful  for  reference : 

^^=(2«-l)/'..,(^)+(2«-5)^.-,W+(2«-9)^...W+....(l) 

—dx -d^  =  (2«  +  ')^»W  (2) 

1 

Spix)dx  =  --i_[/'„_,(^)  -  />.+,(*)]•  (3) 


Art.  19.]  spherical  harmonics.  213 

Art.  18.    Spherical  Harmonics. 

In  problems  in  Potential  where  the  value  of  f^is  given  on  the 
surface  of  a  sphere,  but  is  not  independent  of  the  angle  0,  we 
have  to  solve  Laplace's  Equation  in  the  form  (5),  Art.  i,  and 
by  a  treatment  analogous  to  that  given  in  Art.  4  it  can  be 
proved  that 

V  =r*^  cos  nd)  sin"  6—,  and    V  =  r"^  sm  neb  sm**  6 — j-^-^, 

d^n  dfjT 

where  ^  =  cos  6,  are  particular  solutions  of  (5),  Art.  i. 

The  factors  multiplied  by  r*"  in  these  values  are  known  as 
Tesseral  Harmonics.  They  are  functions  of  0  and  6,  and  they 
play  nearly  the  same  part  in  unsymmetrical  problems  that  the 
Zonal  Harmonics  play  in  those  independent  of  0. 

Y„,{pi,  cp)  =  A^PJ^fx)  +  T  {A^  cos  n(l>  +  B^  sin  «0)sin''  ^^4^ 
is  known  as  a  Surface  Spherical  Harmonic  of  the  mth.  degree, 
and  V  =  r-  Y^{^,  0)    and    F  =  ^  Y^{,u,  0) 

satisfy  Laplace's  Equation,  (5),  Art.  I. 

The  Tesseral  and  the  Zonal  Harmonics  are  special  cases  of 
the  Spherical  Harmonic,  as  is  also  a  form  /*^(cos  y)  known  as 
a  Laplace's  Coefficient  or  a  Laplacian  ;  y  standing  for  the  angle 
between  r  and  the  radius  vector  r,  of  some  fixed  point. 

For  the  properties  and  uses  of  Spherical  Harmonics  we 
refer  the  student  to  more  extended  treatises,  namely,  to 
Ferrer's  Spherical  Harmonics,  to  Heine's  Kugelfunctionen,  or 
to  Byerly's  Fourier's  Series  and  Spherical  Harmonics. 

Art.  19.*    Bessel's  Functions.    Properties. 
We  have  seen  in  Art.  5  that  z  =  J^x)  where 


* 


The  student  should  review  Art.  5  before  reading  this  article. 


214  HARMONIC    FUNCTIONS.  [ChaP.  V. 

is  a  solution  of  the  equation 

^'•^  _l_  I  ^  _|_      —  O  •  (2\ 

dx^      X  dx 

and  we  have  called  JJ^x)  a  Bessel's  Function  or  Cylindrical 
Harmonic  of  the  zero  order. 

/(^)-_'fe  =  fr,_^  ,  ._^ £l_  +  ..1(3) 

■''^'^'~        dx         2_        2. 4^2. 4'. 6      2.4'.ff.8^      J^^-* 
is  called  a  Bessel's  Function  of  the  first  order,  and 

is  a  solution  of  the  equation 


d'^z'   ^    ids'       (  I  \  ,  ,  , 

-dx^+-x-dx+V-^>='^>  (4) 

which  is  the  result  of  differentiating  (2)  with  respect  to  x. 

A  table  giving  values  of  J^x)  and  J^x)  will  be  found  at 
the  end  of  this  chapter. 

If  we  write  JJ^x)  for  s  in  equation  (2),  then  multiply 
through  by  xdx  and  integrate  from  zero  to  x^  simplifying  the 
resulting  equation  by  integration  by  parts,  we  get 

0 

or,  smce  J^{x)  =  —  — ^-, 
dx 

JxJSx)dx  =  xJlx\  (5) 


0 


If  we  write /,(;ir)  for  z  in  equation  (2),  then  multiply  through 
by  ^'— j — K  and  integrate  from  zero  to  x,  simplifying  by  inte- 
gration by  parts,  we  get 

ff('^y+(/.(-^)y]-A/.w)v.=o, 


01 


fx{Jlx)ydx  =  f  [(/.(^))'  +  (/.W)'1.  (6) 

0  .  — ' 


Art  30.]  applications  of  bessel's  functions.  215 

If  we  replace  x  by  jux  in  (2)  it  becomes 

(See  (8),  Art.  5).     Hence  2  =  Jj^^x)  is  a  solution  of  (7). 

If  we  substitute  in  turn  in  {j)  JJ^iux)  and  J^l^hx)  for  2,  mul- 
tiply the  first  equation  by  xJJ^fA.x),  the  second  by  xJl^jx^x), 
subtract  the  second  from  the  first,  simplify  by  integration  by 
parts,  and  reduce,  we  get 

J  xJ^{^^x)J^{}x,x)dx 
0 

T 


Hence  if  ^^  and  /^^  are  different  roots  of  JJ<^^a)  =  o,  or  of 
JXf^^)  =  O,  or  of  ^a/,{^a)  —  \J,{fxa)  —  o, 

a 

J  xj Xf-ux)j Xl^,x)dx  =  o.  (9> 

0 

We    give   without   demonstration   the   following  formulas, 
which  are  sometimes  useful  : 

■n 

JXx)  =  -  /cos  {x  COS  (p)d(f),  (10) 

0 

J^{x)  =  -  I  sin'  0  cos  {x  cos  (p)d4>.  (i  i)- 

7t*/ 


7tt, 
0 


They  can  be  confirmed  by  developing  cos  (x  cos  0),  inte- 
grating, and  comparing  with  (i)  and  (3). 

Art.  20.    Applications  of  Bessel's  Functions. 

(a)  The  problem  of  Art.  5  is  a  special  case  of  the  following : 
The  convex  surface  and  one  base  of  a  cylinder  of  radius  a 
and  length  h  are  kept  at  the  constant  temperature  zero,  the 
temperature  at  each  point  of  the  other  base  is  a  given  function 
of  the  distance  of  the  point  from  the  center  of  the  base ;  re- 


216  HARMONIC    FUNCTIONS.  [ChaP.  V. 

quired  the  temperature  of  any  point  of  the  cylinder  after  the 
permanent  temperatures  have  been  established. 

Here  we  have  to  solve  Laplace's  Equation  in  the  form 

(see  Art.  5),  subject  to  the  conditions 

u  =  o  wheii  ^  =  o, 

u  =  o      ''      r  =  ay 

u=/{r)  ''      z  =  b. 
Starting  with  the  particular  solution  of  (i), 

u  —  ?:m\\{}xz)J,{}xr\  .      (2) 

and  proceeding  as  in  Art.  5,  we  get,  if  yUj ,  /^j,  //g,  .  .  .  are  roots 
of  Jlf^a)  =  0,  (3) 

and        fir)  =  AJi^^r)  +  AJ^u^r)  +  AJl^/)  +  .  .  . ,      (4) 

sinh(/^,^)   .  (      N    ,   ^  gJUJ^Ofag)  / (,,  r\  4-  (0 

{b)  If  instead  of  keeping  the  convex  surface  of  the  cylinder 
at  temperature  zero  we  surround  it  by  a  jacket  impervious  to 
heat  the  equation  of  condition,  u  ^  o  when  r  =  a,  will  be  re- 
placed by  —  =  O  when  r  =  a,  or  ii.  2/  =  sinh  {fAz)JXiJ.r)  by 

^^  =  0     whenr  =  ^, 
dr 

that  is,  by  —  ^J^^a)  =  o 

or  Ji{f^^)  =  o.  (6) 

If  now  in  (4)  and  (5)  //^ ,  yw^,  yUg,  .  .  .  are  roots  of  (6),  (5)  will 
be  the  solution  of  our  new  problem. 

{c)  If  instead  of  keeping  the  convex  surface  of  the  cylinder 
at  the  temperature  zero  we  allow  it  to  cool  in  air  which  is  at 
the  temperature  zero,  the  condition  u  =  o  when  r  =  a  will  be 

replaced  by  -^; — [- Au  =  o  when  r  =  a,  A  being  the  coefficient 

of  surface  conductivity. 


Art.  21.]     development  in  terms  of  bessel's  functions.       217, 

li  u  =  sinh  {}AZ)JJ^ixr)  this  condition  becomes 

—  ^Jx{^r)  +  h/X^r)  =  o    when  r  =  ^, 
or  ^aj,{^a)  —  ah/lfxd)  =  o.  (7) 

If  now  in  (4)  and  (5)  /(, ,  /i, ,  //g ,  .  .  .  are  roots  of  (7),  (5)  will 
be  the  solution  of  our  present  problem. 

It  can  be  shown  that 

/oW  =  o,  (8) 

/,(^)  =  o,  (9) 

and  -^/iW  —  VoW  =  0  (10) 

have  each  an  infinite  number  of  real  positive  roots.*  The 
■earlier  roots  of  these  equations  can  be  obtained  without  serious 
difficulty  from  the  table  for/„(;i:)  and /j(;r)  at  the  end  of  this 
chapter. 

Art.  21.    Development  in  Terms  of  Bessel's  Functions. 

We  shall  now  obtain  the  developments  called  for  in  the  last 
article. 

Let     /{r,  =  AJi^y)  +  AJlu^r)  +  AJIias)  +  .  .  .      (i) 

M,y  fJi^y  /f, ,  etc.,  being  roots  of  /.(f^a)  =  o,  or  of  /,(yw^)  =  o,  or 
of  M^/iiM^)  —  Vo(/^^0  =  O. 

To  determine  any  coefficient  Ak  multiply  (i)  by  rJX^kr)dr 
and  integrate  from  zero  to  a.     The  first  member  will  become 

a 

frAr)Jl,x,r)dr. 

0 

Every  term  of  the  second  member  will  vanish  by  (9),  Art. 
19,  except  the  term 

a 
0 

JrUXl^.r)-\^dr  =  -^,fx[/Xx)-\'dx  =  |  ([/.(Ml'+C/.C/'*^)]') 

0  0 

by  (6),  Art.  19. 

*  See  Riemann's  Partielle  Diflferentialgleichungen,  §  97. 


218  HARMONIC    FUNCTIONS.  [ChaP.  V. 

a 

Hence  A,  =  ^  ^.,.frArV.{M.r)dr,  (2) 

The  development  (i)  holds  good  from  r  =  o  to  r  =  a  (see 
Arts.  6  and  15). 

If  yu,,  yw^,  M,,  etc.,  are  roots  oi /^(jua)  =  O,  (2)  reduces  to 


Ak-- 

a- 


^j^^JrAr)Jl,,r)dr.  (3) 

If  /^i ,  /^3 »  /^3 »  etc.,  are  roots  oi  JJ^i^ia)  —  o,  (2)  reduces  to 

n  /^i,  //2,   f^^y   etc.,   are    roots  of  fxaj,{f.ia)  —  KJXP-d)  —  O,. 
(2)  reduces  to 

For  the  important  case  where /"(r)  =  i 
frA^)Jl^kr)dr=  I  rJl^kr)dr^-~-:,  fxJXx)dx  —  --Jlfx^d)  (6) 

0  0  0 

by  (5),  Art.  19;  and  (3)  reduces  to 

(4)  reduces  to 

A,  -  o,  (8> 

except  ior  k  =  i,  when  yw^  =  o,  and  we  have 

A,  =  l;  (9). 

2A 

(5)  reduces  to         Ak  =  j:r^—. ,   ,.j., r.  (iO> 

Prob.  28.  A  cylinder  of  radius  one  meter  and  altitude  one  meter 
has  its  upper  surface  kept  at  the  temperature  100°,  and  its  base  and 
convex  surface  at  the  temperature  15°,  until  the  stationary  temper- 
atures are  established.  Find  the  temperature  at  points  on  the  axis. 
25,  50,  and  75  centimeters  from  the  base,  and  also  at  a  point  25, 
centimeters  from  the  base  and  50  centimeters  from  the  axis. 

Ans.  29°.6;  47°.6  ;  7i°.2  ;  25°.8 


Art.  ^1.]        DEVELOPMENT    IN    TERMS    OF    BESSEL'S    FUNCTIONS.         219^ 

Prob.  29.  An  iron  cylinder  one  meter  long  and  20  centimeters 
in  diameter  has  its  convex  surface  covered  with  a  so-called  non-con- 
ducting cement  one  centimeter  thick.  One  end  and  the  convex 
surface  of  the  cylinder  thus  coated  are  kept  at  the  temperature  zero, 
the  other  end  at  the  temperature  of  100  degrees.  Given  that  the  con- 
ductivity of  iron  is  0.185  ^^^  ^^  cement  0.000162  in  C.  G.  S.  units. 

Find  to  the  nearest  tenth  of  a  degree  the  temperature  of  the  mid- 
dle point  of  the  axis,  and  of  the  points  of  the  axis  20  centimeters 
from  each  end  after  the  temperatures  have  ceased  to  change. 

Find  also  the  temperature  of  a  point  on  the  surface  midway  be- 
tween the  ends,  and  of  points  of  the  surface  20  centimeters  from 
each  end.  Find  the  temperatures  of  the  three  points  of  the  axis, 
supposing  the  coating  a  perfect  non-conductor,  and  again,  suppos- 
ing the  coating  absent.  Neglect  the  curvature  of  the  coating.  Ans. 
i5°.4  ;  40°.85  ;  72°.8  ;  i5°.3  ;  40°.;  ;  72°.5  ;  o°.o  ;  o°.o  ;  1°.^- 

Prob.  30.  If  the  temperature  at  any  point  in  an  infinitely  long 
cylinder  of  radius  c  is  initially  a  function  of  the  distance  of  the 
point  from  the  axis,  the  temperature  at  any  time  must  satisfy  the 

du         o  id'^u    ,    I  du\   ,         .  .      .         .    .      ,      ,    . 

equation    ~  —  a    (  — ^  -i —  —  1   (see  Art.   i),  since  it  is  clearly  in- 
dependent of  z  and  0. 
Show  that 

where,  if  the  surface  of  the  cylinder  is  kept  at  the  temperature 
zero,  )Uj ,  //, ,  //, , . . .  are  roots  of  jSl^^)  =  o  and  Ak  is  the  value 
given  in  (3)  with  c  written  in  place  o(  a  ;  if  the  surface  of  the  cylin- 
der is  adiabatic  M^,  Ma  Mit  '  ■  •  are  roots  of  /^{f^c)  =  o  and  Aj^,  is  ob- 
tained from  (4);  and  if  heat  escapes  at  the  surface  into  air  at  the  tem- 
perature zero  //,,  //,,  //,,...  are  roots  of  )^cjX^c)  —  ^/Xl^^)  —  ^^ 
and  Ak  is  obtained  from  (5). 

Prob.  31.  If  the  cylinder  described  in  problem  29  is  very  long 
and  is  initially  at  the  temperature  100°  throughout,  and  the  con- 
vex surface  is  kept  at  the  temperature  0°,  find  the  temperature  of  a 
point  5  centimeters  from  the  axis  15  minutes  after  cooling  has  begun  ; 
first  when  the  cylinder  is  coated,  and  second,  when  the  coating  is 
absent.     Ans.   97°. 2  ;  o°.oi. 

Prob.  32.  A  circular  drumhead  of  radius  a  is  initially  slightly 
distorted  into  a  given  form  which  is  a  surface  of  revolution  about 
the  axis  of  the  drum,  and  is  then  allowed  to  vibrate,  and  z  is  the 
ordinate  of  any  point  of  the  membrane  at  any  time.    Assuming  that 


220  HARMONIC    FUNCTIONS.  [Chap.  V. 

.  r     -,  .     d^z        Jd*z      1  dz\      ^. 

z  must  satisfy  the  equation  ^  =  c  [^  +  —  o:^  i,  subject  to  the  con- 

'dz 
ditions  -?  =  o  when  r  —  a,  —-  =^  o  when  /  =  o,  and  s"  =  fir)  when 

ot 

/  =  o,  show  that  z  —  AJJ^^x^r)  cos  p.,ct  +  A^JX^^^)  cos  ^^ct  +  .  . . 
where  >"i,  /^,,  /^,, .  •  •  are  roots  of  /o(/^)  =  o  and  ^^  has  the  value 
given  in  (3). 

Prob.  33.  Show  that  if  a  drumhead  be  initially  distorted  as  in 
problem  32  it  will  not  in  general  give  a  musical  note  ;  that  it  may  be 
initially  distorted  so  as  to  give  a  musical  note  ;  that  in  this  case  the 
vibration  will  be  a  steady  vibration  ;  that  the  periods  of  the  various 
musical  notes  that  can  be  given  are  proportional  to  the  roots  of 
Jj^x)  =  o,  and  that  the  possible  nodal  lines  for  such  vibrations 
are  concentric  circles  whose  radii  are  proportional  to  the  roots  of 

Art.  22.    Problems  in  Bessel's  Functions. 

If  in  a  problem  on  the  stationary  temperatures  of  a  cylinder 
u  =  o  when  :s  =  o,  u  =  o  when  z^  b,  and  u  =  f{z)  when  r  =  a, 
the  problem  is  easily  solved.  If  in  (2),  Art.  20,  and  in  the  cor- 
responding solution  3  =  cosh  (/^z)/X/ir)  we  replace  ju  by  yu?,  we 
•can  readily  obtain  2  =  sin  (fxz)JXiAri)  and  z  =  cos  (M^)/o(M^i) 
as  particular  solutions  of  (i),  Art.  20;  and 


and  is  real. 


y„(^0=l+^  +  ^  +  5r^+...  (1) 


fc=l 


km 


where  Aje^-j-  I  f{z)s{n-j-  dz  (2) 

0 
by  Art.  9. 

krrri^ 


I  k7rri\ 
k=\  ^     _  /  knai  \ 

is  the  required  solution. 


Hence  „  =  ^  ^,  sin  —  — —  (3) 


Art.  24.]  lame's  functions.  221 

A  table  giving  the  values  of  Jj^xi)  will  be  found  at  the  end 
of  this  chapter. 

Prob.  34.  A  cylinder  two  feet  long  and  two  feet  in  diameter  has 
its  bases  kept  at  the  temperature  zero  and  its  convex  surface  at 
100  degrees  Centigrade  until  the  internal  temperatures  have  ceased 
to  change.  Find  the  temperature  of  a  point  on  the  axis  half  way 
between  the  bases,  and  of  a  point  six  inches  from  the  axis,  half  way 
between  the  bases.     Ans.  72.°!;  8o°.i. 

Art.  23.    Bessel's  Functions  of  Higher  Order. 

If  we  are  dealing  with  Laplace's  Equation  in  Cylindrical 
Coordinates  and  the  problem  is  not  symmetrical  about  an 
axis,  functions  of  the  form 

x'  ,  X' 


^^^""^^Ynxk^h 


2''I\n  +  i)L  2\7t  +1)  '  2\2\{n-\-  i){n  +  2) 
play  very  much  the  same  part  as  that  played  hy  JJ^x)  in  the 
preceding  articles.  They  are  known  as  Bessel's  Functions  of 
the  nth.  order.  In  problems  concerning  hollow  cylinders  much 
more  complicated  functions  enter,  known  as  Bessel's  Functions 
of  the  second  kind. 

For  a  very  brief  discussion  of  these  functions  the  reader  is 
referred  to  Byerly's  Fourier's  Series  and  Spherical  Harmonics  ; 
for  a  much  more  complete  treatment  to  Gray  and  Matthews* 
admirable  treatise  on  Bessel's  Functions. 

Art.  24.    LAMiE's  Functions. 

Complicated  problems  in  Potential  and  in  allied  subjects  are 
usually  handled  by  the  aid  of  various  forms  of  curvilinear  co- 
ordinates, and  each  form  has  its  appropriate  Harmonic  Func- 
tions, which  are  usually  extremely  complicated.  For  instance, 
Lame's  Functions  or  Ellipsoidal  Harmonics  are  used  when 
solutions  of  Laplace's  Equation  in  Ellipsoidal  coordinates  are 
required ;  Toroidal  Harmonics  when  solutions  of  Laplace's 
Equation  in  Toroidal  coordinates  are  needed. 

For  a  brief  introduction  to  the  theory  of  these  functions 
see  Byerly's  Fourier's  Series  and  Spherical  Harmonics.    ' 


222 


HARMONIC    FUNCTIONS. 


[Chap.  V. 


Table  I.  Surface  Zonal  Harmonics. 

0 

P,  (cos  6) 

P, (cos  0) 

Pg  (COS  0) 

P4  (cos  0) 

P5  (cos  0) 

Pg  (COS  0) 

P7  (cos  0) 

0' 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1 

.9998 

.9995 

.9991 

.9985 

.9977 

.9967 

.9955 

2 

.9994 

.9982 

.9963 

.9939 

.9909 

.9872 

.9829 

3 

.9986 

.9959 

.9918 

.9863 

.9795 

.9713 

.9617 

4 

9976 

.9927 

.9854 

.9758 

.9638 

.9495 

.9329 

5 

.9962 

.9886 

.9773 

.9623 

.9437 

.9216 

.8961 

6 

.9945 

.9836 

.9674 

.9459 

.9194 

.8881 

.8522 

7 

.99.25 

.9777 

.9557 

.9267 

.8911 

.8476 

.7986 

8 

.  9903 

.9709 

.9423 

.9048 

.8589 

.8053 

.7448 

9 

.9877 

.9633 

.9273 

.8803 

.8232 

.7571 

.6831 

10 

.9848 

.9548 

.9106 

.8532 

.7840 

.7045 

.6164 

11 

.9816 

.  9454 

.8923 

.8238 

.7417 

.6483 

.0461 

12 

.9781 

.9352 

.8724 

.7920 

.6966 

.5892 

.4732 

13 

.9744 

.9241 

.8511 

.7582 

.6489 

.5273 

.3940 

14 

.9703 

.9122 

.8283 

.7224 

.5990 

.4635 

.3219 

15 

.9659 

.8995 

.8042 

.6847 

.5471 

.3982 

.2J54 

16 

.9613 

.8860 

.7787 

.6454 

.4937 

.3322 

.1699 

17 

.9563 

.8718 

.7519 

.6046 

.4391 

.2660 

.0961 

18 

.9511 

.8568 

.7240 

.5624 

.3836 

.2002 

.028.') 

19 

.9455 

.8410 

.6950 

.5192 

.3276 

.1347 

-.0443 

20 

.9397 

.8245 

.6649 

.4750 

.2715 

.0719 

-.1072 

21 

.9336 

.8074 

.6338 

.4300 

.2156 

.0107 

-.1662 

22 

.9272 

.7895 

.6019 

.3845 

.1602 

-.0481 

-.2201 

23 

.9205 

.7710 

.5692 

.3386 

.1057 

-.1038 

-.2681 

24 

.9135 

.7518 

.5357 

.2926 

.0525 

-.1559 

-.3095 

25 

.9063 

.7321 

.5016 

.2465 

.0009 

-.2053 

-.3463 

26 

.8988 

.7117 

.4670 

.2007 

-.0489 

-.2478 

-.3717 

■27 

.8910 

.6908 

.4319 

.1553 

-.0964 

-.2869 

-.3921 

.28 

.8829 

.6694 

.3964 

.1105 

-.1415 

-.3211 

-.4052 

29 

.8746 

.6474 

.3607 

.0665 

-.1839 

-.3503 

-.4114 

:30 

.8660 

.6250 

.3248 

.0234 

-.2233 

-.3740 

-.4101 

31 

.8572 

.6021 

.2887 

-.0185 

-.2595 

-.3924 

-.4022 

32 

.8480 

.5788 

.2527 

-.0591 

-.2923 

-.4052 

-.3876 

33 

.8387 

.5551 

.2167 

-.0982 

-.3216 

-.4126 

-.3670 

34 

.8290 

.5310 

.1809 

-.1357 

-.3473 

-.4148 

-.3409 

35 

.8192 

.5065 

.1454 

-.1714 

-.3691 

-.4115 

-.3096 

36 

.8090 

.4818 

.1102 

-.2052 

-.3871 

-.4031 

-.2738 

37 

.7986 

.4567 

.0755 

-.2370 

-.4011 

-.3898 

-.2343 

•38 

.7880 

.4314 

.0413 

-.2666 

-.4112 

-.3719 

-.1018 

39 

.7771 

.4059 

.0077 

-.2940 

-.4174 

-.3497 

-.1469 

40 

.7660 

.3802 

-.0252 

-.3190 

-.4197 

-.3234 

-.1003 

41 

.7547 

.3544 

-.0574 

-.3416 

-.4181 

-.2938 

-.0534 

42 

.7431 

.3284 

-.0887 

-.3616 

-.4128 

-.2611 

-.0065 

43 

.7314 

.3023 

-.1191 

-.3791 

-.4038 

-.2255 

.0398 

44 

.7193 

.2762 

-.1485 

-.3940 

-.3914 

-.1878 

.0846 

45° 

.7071 

.2500 

-.1768 

-.4062 

-.3757 

-.1485 

.1270 

TABLES. 


223 


Table  I.    Surface  Zonal  Harmonics. 


e 

Pi  (cos  6) 

Pa  cos  B) 

IPs  (cos  0)  j 

P4  (cos  6) 

Pfi  (cos  0) 

Pe  (cos  9) 

P7  (cos  0) 

45° 

.7071 

.2500 

-.1768 ; 

-.4062 

-.3757 

-.1485 

.1270 

46 

.6947 

.2238 

-.2040 

-.4158 

-.3568 

-.1079 

.1666 

47 

.6820 

.1977 

-.2300 

-.4252 

-.3350 

-.0645 

.2054 

48 

.6691 

.1716 

-.2547  i 

-.4270 

-.3105 

-.0251 

.2349 

49 

.6561 

.1456 

-.2781 

-.4286 

-.2836 

.0161 

.2627 

50 

.6428 

.1198 

-.3002 

-.4275 

-.2545 

.0563 

.2854 

51 

.6293 

.0941 

-.3209  1 

-.4239 

-.2235 

.0954 

.3031 

52 

.6157 

.0686 

-.3401 

-.4178 

-.1910 

.1326 

.3153 

53 

.6018 

.0433 

-.3578  1 

-.4093 

-.1571 

.1677 

.3221 

54 

.5878 

.0182 

-.3740 

-.3984 

-.1223 

.2002 

.3234 

55 

.5733 

-.0065 

-.3886 

-.3852 

-.0868 

.2297 

.3191 

56 

.5592 

-.0310 

-.4016 

-.3698 

-.0510 

.2559 

.3095 

57 

.5446 

-.0551 

-.4131 

-.3524 

-.0150 

.2787 

.2949 

58 

.5299 

-.07-8 

-.4229 

-.3331 

.0206 

.2976 

.2752 

59 

.5150 

-.1021 

-.4310 

-.3119 

.0557 

.3125 

.2511 

m 

.5000 

-.1250 

-.4375 

-.2891 

.0898 

.3232 

.2231 

«i 

.4848 

-.1474 

-.4423 

-.2647 

.1229 

.8298 

.1916 

4J2 

.4695 

-.1694 

-.4455 

-.2390 

.1545 

.3321 

.1571 

m 

.4540 

-.1908 

-.4471 

-.2121 

.1844 

.3302 

.1203 

u 

.4384 

-.2117 

-.4470 

-.1841 

.2123 

.3240 

.0818 

<55 

.4226 

-.2321 

-.4462 

-.1552 

.2381 

.3138 

.0422 

66 

.4007 

-.•i518 

-.4419 

-.1256 

.2615 

.2996 

.0021 

<$7 

.3907 

-.2710 

-.4370 

-.0955 

.2824 

.2819 

-.0375 

68 

.3746 

-.2896 

-.4305 

-.0650 

.3005 

.2605 

-.0763 

«9 

.3584 

-.3074 

-.4225 

-.0344 

.3158 

.2361 

-.1135 

70 

.3420 

-.3245 

-.4130 

-.0038 

.3281 

.2089 

-  1485 

71 

.3256 

-.3410 

-.4021 

.0267 

.3373 

.1786 

-.1811 

72 

.3090 

-.3568 

-.3898 

.0568 

.3434 

.1472 

-.2099 

73 

.2924 

-.3718 

-.3761 

.0864 

.3463 

.1144 

-.2347 

74 

.27r)0 

-.3860 

-.3611 

.1153 

.3461 

.0795 

-.2559 

75 

.2588 

-.3995 

-.3149 

.1434 

.3427 

.0431 

-.2730 

76 

.2419 

-.4112 

-.3275 

.1705 

.3362 

.0076 

-.2848 

77 

.2250 

-.4241 

-.3090 

.1964 

.3267 

-.0284 

-.2919 

78 

.':079 

-.4352 

-.2894 

.2211 

.3143 

-.0644 

-.2943 

79 

.1908 

-.4454 

-.2688 

.2443 

.2990 

-.0989 

-.2913 

■80 

.1736 

-.4548 

-.2474 

.2659 

.2810 

-.1321 

-.2835 

81 

.1564 

-.4633 

-.2251 

.2859 

.2606 

-.1635 

-.2709 

82 

.1392 

-.4709 

-.2020 

.3040 

.2378 

-.1926 

-.2536 

83 

.1219 

-.4777 

-.1783 

.3203 

.2129 

-.2193 

-.2321 

84 

.1045 

-.4836 

-.1539 

.3345 

.1861 

-.2431 

-.2067 

85 

.0872 

-.4886 

-.1291 

.3468 

.1577 

-.2638 

-.1779 

86 

.0698 

-.4927 

-.1038 

.3569 

.1278 

-.2811 

-.1460 

87 

.0523 

-.4959 

-.0781 

.3648 

.0969 

-.2947 

-.1117 

88 

.0349 

-.4982 

-.0522 

.3704 

.0651 

-.3045 

-.0735 

89 

.0175 

-.4995 

-.0262 

.3739 

.0327 

-.3105 

-.0381 

90' 

.0000 

-.5000 

.0000 

.3750 

.0000 

-.3125 

.0000 

224 


HARMONIC    FUNCTIONS. 


[Chap.  V. 


Table  II.    Bkssel's   Functions. 


X 

Joix) 

Ji(x)      j 

X 

Jo(x) 

Jiix)       \ 

^  t 

Joix) 

Jiix) 

0.0 

1.0000 

0.0000 

5.0 

-.1776 

-.3276  1 

10.0  i 

-.2459 

.0435 

0.1 

.9975 

.0499  1 

5.1 

-.1443 

-.8871 : 

10.1  1 

-.2490 

.0184 

0.2 

.9900 

.0995  1 

5.2 

-.1108 

-.8482  1 

10.2 

-.2490 

.0066 

0.3 

.9776 

.1483  : 

5.8 

-.0758 

-.3460 

10.3 

-.2477 

-.0818 

0.4 

.9604 

.1960  1 

5.4 

-.0412 

-.3453 

10.4 

-.2484 

-.0555 

0.5 

.9385 

.2423  ' 

5.5 

-.0068 

-.8414 

10.5 

-.2866 

-.0789 

0.6 

.9120 

.2867  ' 

5.6 

.0270 

-.8848  1 

10.6 

-.2276 

-.1012 

0.7 

.8812 

.8290  1 

5.7 

.0599 

-.3241 

10.7 

-.2164 

-.1224 

0.8 

.8463 

.3688  ' 

5.8 

.0917 

-.3110 

10.8 

-.2082 

-.1422 

0.9 

.8075 

.4060  1 

5.9 

.1220 

-.2951 

10.9 

-.1881 

-.1604 

1.0 

.7652 

.4401  ' 

6.0 

.1506 

-.2767 

11.0 

-.1712 

-.1768 

1.1 

.7196 

.4709 

61 

.1778 

-.2559 

11.1 

-.1528 

-.1913 

1.2 

.6711 

.4988  i 

6.2 

.2017 

-.2329 

11.2 

-.1330 

-.2039 

1.3 

.6201 

.5220  1 

6.8 

.2238 

-.2081 

11.3 

-.1121 

-.2143 

1.4 

.5669 

.5419  j 

6.4 

.2433 

-.1816 

11.4 

-.0902 

-.2225 

1.5 

.5118 

.5579 

6.5 

.2601 

-.1538 

11.5 

-.0677 

-.2284 

1.6 

.4554 

5699 

6.6 

.2740 

-.1250  1 

11.6 

-.0446 

-.2320 

1.7 

.3980 

.5778  , 

6.7 

.2851 

-.0958 

11.7 

-.0213 

-.2883 

1.8 

.8400 

.5815  ' 

6.8 

.2931 

-.0652 

11.8 

.0020 

-.2323 

1.9 

.2818 

.5812  i 

6.9 

.2981 

-.0349 

11.9 

.0250 

-.2290 

55.0 

.2239 

.5767 

7.0 

.8001 

-.0047 

12.0 

.0477 

-.2234 

21 

.1666 

.5683 

7.1 

.2991 

.0252 

12.1 

.0697 

-.2157 

2.2 

.11(J4 

.5560  : 

7.2 

.2951 

.0543 

12.2 

.0908 

-.2(60 

2.3 

.0555 

.5399 

7.3 

.2882 

.0826 

12.3 

.1108 

-.1948 

2.4 

.0025 

.5202 

7.4 

.2786 

.1096 

12.4 

.1296 

-.1807 

2.5 

-.0484 

.4971  , 

7.5 

.2663 

.1352 

12.5 

.1469 

-.1655 

2.6 

-.0968 

.4708 

7.6 

.2516 

.1592 

12.6 

.1626 

-.1487 

2.7 

-.1424 

.4416 

7.7 

.2846 

.1813 

12.7 

.1766 

-.1307 

2.8 

-.1850 

.4007 

7.8 

.2154 

.2014 

-12.8 

.1887 

-.1114 

2.9 

-.2243 

.8754 

79 

.1944 

.2193 

12.9 

.1988 

-.0912 

8.0 

-.2601 

.8891 

8.0 

.1717 

.2346 

13.0 

.2069 

-.0708 

3.1 

-.2!)21 

.8009 

8.1 

.1475 

.2476 

131 

.2129 

-.0489 

3.2 

-.3202 

.2618 

82 

.1222 

.2580 

13.2 

.2167 

-.0271 

3.3 

-.3443 

.2207 

8.3 

.0960 

.2657 

13.3 

.2183 

-.0052 

3.4 

-.8643 

.1792 

8.4 

.0692 

.2708 

13.4 

.2177 

.01(i6 

3.5 

-.8801 

.1874 

8.5 

.0419 

.2731 

13.5 

.2150 

.0380 

3.6 

-.8918 

,095-) 

8.6 

.0146 

.2728 

13.6 

.2101 

.0590 

3.7 

-.8992 

.0588 

8.7 

-.0125 

.2697 

13.7 

.2032 

.0791 

8.8 

-.4026 

.0128 

8.8 

-.0392 

.2641 

13.8 

.1943 

.0984 

3.9 

-.4018 

-.0272 

8.9 

-.0653 

.2559 

18.9 

.1886 

.1166 

4.0 

-.3972 

-.0660 

9.0 

-.0903 

.2458 

14.0 

.1711 

.1884 

4.1 

-.3887 

-.1088 

9.1 

-.1142 

.2824 

14.1 

.1570 

.1488 

4.2^ 

-.3766 

-.1886 

9.2 

-.1367 

.2174 

14.2 

.1414 

.1626 

4.3 

-.8610 

-.1719 

9.3 

-.1577 

.2004 

14.3 

.1245 

.1747 

4.4 

-.8423 

-.2028 

9.4 

-.1768 

.1816 

14.4 

.1065 

.1850 

4.5 

-.8205 

-.2811 

9.5 

-.1989 

.1613 

14.5 

.0875 

.1984 

4.6 

-.2961 

-.2566 

9.6 

-  .2090 

.1895 

14.6 

.0679 

.1999 

4.7 

-.2698 

-.2791 

9.7 

-.2218 

.1166 

14.7 

.0476 

.2048 

4.8 

-.2404 

-.2985 

9.8 

-.2828 

.0928 

14.8 

.0271 

.2066 

4.9 

-.2097 

-.8147 

9.9 

-.2403 

.0684 

14.9 

.0064 

.20!i9 

5.0 

-.1776 

-.3276 

10.0 

-.2459 

.0485 

15.0 

-.0142 

.2051 

TABLES.  225 

Table  III.— Roots  of  Bessel's  Functions. 


n 

a;„  for  Jo(x„)  =  0 

x^  for  Ji(x„)  =  0 

n 

a;„  for  Joix,^)  =  0 

x„  for  Jiix„)  =  0 

1 
2 
3 
4 
5 

2.4048 

5.5201 

8.6537 

11.7915 

14.9309 

3.8317 

7.0156 

10.1735 

13.3237 

16.4706 

6 

7 

8 

9 

10 

18.0711 
21.2116 
24.3525 
27.4935 
30.6346 

19.6159 
22.7601 
25  9037 
29.0468 
32.1897 

Table  IV.— Values  of  Jo{xi). 


X 

JoCxi) 

X 

Joixi) 

X 

Joioci) 

0.0 

1.0000 

3.0 

3.2796 

4.0 

11.3019 

0.1 

1.0025 

3.1 

2.4463 

4.1 

12.3236 

0.2 

1.0100 

3.3 

2.6291 

4.2 

13.4425 

0.3 

1.0226 

3.3 

2.8296 

4.3 

14.6680 

0.4 

1.0404 

2.4 

3.0493 

4.4 

16.0104 

0.5 

1.0635 

3.5 

3.2898 

4.5 

17.4813 

0.6 

1.0920 

3.6 

3.5533 

4.6 

19.0926 

0.7 

1.1263 

3.7 

3.8417 

4.7 

20.8585 

0.8 

1 . 1665 

3.8 

4.1573 

4.8 

22.7937 

0.9 

1.2130 

3.9 

4.5027 

4.9 

24  914& 

1.0 

1.2661 

3.0 

4.8808 

5.0 

27.2399- 

1.1 

1.3262 

3.1 

5.2945 

5.1 

29.7889 

1.2 

1.3937 

3.3 

5.7472 

5.2 

32.5836 

1.3 

1.4963 

3.3 

6.2426 

5.3 

35.6481 

1.4 

1.5534 

3.4 

6.7848 

5.4 

39.0088 

1.5 

1.6467 

3.5 

7.3782 

6.5 

42.6946 

1.6 

1.7500 

3.6 

8.0277 

5.6 

46.7376 

1.7 

1.8640 

3.7 

8.7386 

5.7 

51.1725 

1.8 

1.9896 

3.8 

9.5169 

5.8 

56.0381 

1.9 

3.1377 

3.9 

10.3690 

5.9 

61.3766 

236  FUNCTIONS  OF   A   COMPLEX   VARIABLE.  [ChAP.  VL 


Chapter  VI. 

FUNCTIONS  OF  A  COMPLEX  VARIABLE. 

By  Thomas  S.  Fiske, 
Adjunct  Professor  of  Mathematics  in  Columbia  University. 

Art.  L    Definition  of  Function. 

If  two  or  more  quantities  are  such  that  no  one  of  them, 
when  any  values  whatsoever  are  assigned  to  the^others,  suf- 
fers any  restriction  in  regard  to  the  values  which  it  can  assume 
the  quantities  are  said  to  be  *'  independent." 

If  one  quantity  is  so  related  to  another  quantity  or  to 
several  independent  quantities,  that  whenever  particular  values 
are  assigned  to  the  latter,  the  former  is  required  to  take  one  or 
another  of  a  system  of  completely  determined  values,  the  for- 
mer is  said  to  be  a  "  function  "  of  the  latter.  The  quantity  or 
quantities  upon  the  values  of  which  the  value  of  the  function 
depends,  are  said  to  be  the  "  independent  variables  "  of  the 
function. 

A  function  is  "  one-valued  "  when  to  every  set  of  values  as- 
signed to  the  independent  variables  there  corresponds  but  one 
value  of  the  function.  It  is  said  to  be  "  ^^-valued  "  when  to 
every  set  of  values  of  the  independent  variables  n  values  of  the 
function  correspond. 

The  **  Theory  of  Functions "  has  among  its  objects  the 
study  of  the  properties  of  functions,  their  classification  accord- 
ing to  their  properties,  the  derivation  of  formulas  which  exhibit 
the  relations  of  functions  to  one  another  or  to  their  independ- 
ent variables,  and  the  determination  whether  or  not  functions 
exist  satisfying  assigned  condition^. 


Art.  2.] 


REPRESENTATION    OF    COMPLEX    VARIABLE. 


227 


Art.  2.    Representation  of  Complex  Variable. 

A  variable  quantity  is  capable,  in  general,  of  assuming  both 
real  and  imaginary  values.  In  fact,  unless  it  be  otherwise 
specified,  every  quantity  w  is  to  be  regarded  as  having  the 
"  complex  "  form  u  +  v    V  —  i,  u  and  v  being  real.     It  is  cus- 


tomary to  denote  V  —  i  by  i,  and  to  write  the  preceding  quan- 
tity thus  :  u  -\-  iv.  If  2^  is  zero,  w^  is  real ;  if  u  is  zero,  zv  is  a 
^*  pure  imaginary." 

A  quantity  z  =:  x  -\-  iy  \s  said  to  vary  "  continuously  "  when 
between  every  pair  of  values  v/hich  it  takes,  c^=^  a^-{-  ib^  and 
c^^=L  a^-\-  ib^ ,  the  value  of  z  varies  in  such  a  manner  that  x  and 
y  pass  through  all  real  values  intermediate  to  a^  and  a^ ,  b^ 
and  ^3,  respectively. 

It  is  usual  to  give  to  a  variable  quantity  z^^  x  -\-iy  2,  graphi- 
cal representation  by  drawing  in  a  plane  a  pair  of  rectangular 
axes  and  constructing  a  point  whose  abscissa  and  ordinate  are 
respectively  equal  to  x  and  y.  To  every  value  of  z  will  corre- 
spond a  point ;  and,  conversely,  to  every  point  will  correspond 
a  value  of  z.  The  terms  *'  point  "  and  value,  then,  may  be  inter- 
changed without  confusion.  When  z  varies  continuously  the 
graphical  representation  of  its  varia- 
tion, or  its  "  path,"  will  be  a  continuous 
line.  This  graphical  representation  is 
of  the  highest  importance.  By  means 
of  it  some  of  the  most  complicated 
propositions  may  be  given  an  exceed- 
ingly condensed  and  concrete  expres- 
sion. 

By  putting  x  =  r  cos  6,  y  =  r  sin  6,  where  r  is  a  positive  real 
quantity,  the  point 

z  =  r(cos  6  -\-i  sin  6) 

is  referred  to  polar  coordinates.  The  quantity  r  is  called  the 
absolute  value  or  "  modulus  "  of  z.  It  is  often  written  \z\  .  6 
is  known  as  the  *'  argument  "  of  z. 


228  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [ChaP.  VL 

A  function  is  sometimes  considered  for  only  such  values  of 
each  independent  variable  as  are  represented  graphically  by  the 
points  of  a  certain  continuous  line.  In  the  study  of  functions 
of  real  variables,  for  example,  the  path  of  each  variable  is  rep- 
resented by  a  straight  line,  namely,  the  axis  of  real  quantities, 
or  J  =  o. 

Art.  3.    Absolute  Convergence. 

The  representation  of  functions  by  means  of  infinite  series 
IS  one  of  the  most  important  branches  of  the  theory  of  func- 
tions. In  many  problems,  in  fact,  it  is  only  by  means  of  series 
that  it  is  possible  to  determine  functions  satisfying  the  condi- 
tions assigned  and  to  obtain  the  required  numerical  results. 
Frequent  use  will  be  made  of  the  following  theorem. 

Theorem. — If  the  moduli  of  the  terms  of  a  series  form  a 
convergent  series,  the  given  series  is  convergent. 

Let  the  given  series  be  W=  w^-\-w^-\-  .  .  .  -f-  ^«  + .  •  . 
in  which  w^  =  r^  (cos  0^  -|-  ^  sin  6^),  w,  =  r,  (cos  6,  +  ^'sin  6^,) .  .  . 
By  hypothesis  the  series  R  =  r^ -{-  r^  -\-  .  .  .  -\-  r„ -{-  .  .  .  is 
convergent.  Its  terms  being  all  positive,  the  sum  of  its  first  m 
terms  constantly  increases  with  m,  but  in  such  a  manner  as  to 
approach  a  limit.  The  same  will  be  true  necessarily  of  any 
series  formed  by  selecting  terms  from  R.  The  sum  of  the  first 
m  terms  of  the  series  W  is  composed  of  two  parts, 
r„  cos  6^  +  r,cos6^  .  .  .  +  r^_^  cos  6^.,, 
i{r,  sin  ^„  +  r,  sin  ^,  +  .  .  .  +  r^_^  sin  8,^_,\ 
and  each  of  these  in  turn  may  be  divided  into  parts  which  have 
all  their  terms  of  the  same  sign.  Every  one  of  the  four  parts 
thus  obtained  approaches  a  limit  as  m  is  increased  ;  for  the 
terms  of  each  part  have  the  same  sign,  and  cannot  exceed,  in 
absolute  value,  the  corresponding  terms  of  R.  Hence,  as  m  is 
increased,  the  sum  of  the  first  m  terms  of  W  approaches  a 
limit ;  which  was  to  be  proved. 

A  series,  the  moduli  of  whose  terms  form  a  convergent 
series,  is  said  to  be  *'  absolutely  convergent." 


Art.  4.J  elementary  functions.  229 

Prob.  I.  Show  that  the  series  i  -{•  z -\-  z^  -\- .  .  .  +  2"  -}-  •  •  •  is 
absolutely  convergent,  if    |  2  |  <  i. 

Art.  4.     Elementary  Functions. 

In  elementary  mathematics  the  functions  are  usually  con- 
sidered for  only  real  values  of  the  independent  variables.  In 
the  case  of  the  algebraic  functions,  however,  there  is  no  diffi- 
culty in  assuming  that  the  independent  variables  are  complex. 
The  theory  of  elimination  shows  that  every  algebraic  equation 
can  be  freed  from  radicals.  Every  algebraic  function,  there- 
fore, is  defined  by  an  equation  which  may  be  put  in  a  form 
wherein  the  second  member  is  zero  and  the  first  member  is 
rational  and  entire  in  the  function  and  its  independent  variables. 

Besides  the  algebraic  functions,  the  functions  most  often 
occurring  in  elementary  mathematics  are  the  trigonometric  and 
exponential  functions  and  the  functions  inverse  to  them.  The 
definitions,  by  which  these  functions  are  generally  first  intro- 
duced, have  no  significance  in  the  case  where  the  inde- 
pendent variables  are  complex.  However,  the  following 
familiar  series, 

^  =  exp<^=  1+^+^  +  ^  +  ^+..., 

sm^=^--4--j-  — +  ... 

which  have  been  established  for  the  case  where  the  variables 
are  real,  furnish  most  convenient  general  definitions  for  exp  ^, 
cos  2,  and  sin  2:,  these  series  being  absolutely  convergent  for 
every  finite  value  of  2.  Defining  the  logarithmic  function  by 
the  equation 

^log  z  —  exp  (log  js)  =  jSf 
it  follows  that 

^*  _  ^  loga  _  gxp  (^  log  a). 


230  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VL 

The  following  equations  also  are  to  be  regarded  as  equations 
of  definition : 

sin  <s:  ^  cos^s. 

tan  z  =  ,  cot  z  =  -- — , 

cos  z  sm  z 

I         .                             I 
sec  z  = ,         cosec  z  =  — . 


cos^  sin  <3: 

It  may  be  shown  that  the  formulas  which  are  usually  obtained 
on  the  supposition  that  the  independent  variables  are  real,  and 
which  express  in  that  case  properties  of  and  relations  between 
the  preceding  functions,  still  hold  when  the  independent 
variables  are  complex. 

Prob.  2.  Show  that  ^'V  =  e"^"^  ",  m  and  n  being  complex. 
Prob.  3.   Deduce   cos  z  =  W  +  ^""),  sin  z  —  ~.(e''  —  e'''). 

21 

Prob.  4.  Deduce  cos  (z^  -\-  z^)  =  cos  z^  cos  z^  —  sin  z^  sin  z^ , 
sin  (2;,  +  2J  =  cos  z^  sin  z^  +  sin  z^  cos  z^ . 

Art.  5.    Continuity  of  Functions. 

Let  a  function  of  a  single  independent  variable  have  a 
determinate  value  for  a  given  value  c  of  the  independent  vari- 
able. If,  when  the  independent  variable  is  made  to  approach 
Cj  whatever  supposition  be  made  as  to  the  method  of  approach, 
the  function  approaches  as  a  limit  its  determinate  value  at  c, 
the  function  is  said  to  be  "  continuous  "  at  c. 

This  definition  may  be  otherwise  expressed  as  follows :  A 
function  of  a  single  independent  variable  is  continuous  at  the 
point  c,  when,  being  given  any  positive  quantity  e,  it  is  possible 
to  construct  a  circle,  with  center  at  c  and  radius  equal  to  a 
determinate  quantity  d,  so  small  that  the  modulus  of  the 
difference  between  the  value  of  the  function  at  the  center  and 
that  at  every  other  point  within  the  circle  is  less  than  e. 

A  function  of  several  independent  variables  is  said  to  be 
continuous  for  a  particular  set  of  values  assigned  to  those  vari- 
ables, when  it  takes  for  that  set  of  values  a  determinate  value 
c,  and  for  every  new  set  of  values,  obtained  by   altering  the 


Art.  5.]  continuity  of  functions.  231 

variables  by  quantities  of  moduli  less  than  some  determinate 
positive  quantity  d,  the  value  of  the  function  is  altered  by  a 
qjantity  of  modulus  less  than  any  previously  chosen  arbitrarily 
snail  positive  quantity  e. 

A  function  of  one  independent  variable  is  said  to  be  con- 
tnuous  in  a  given  region  of  the  plane  upon  which  its  indepen- 
dent variable  is  represented,  if  it  is  contiuuous  at  every  point 
in  that  region. 

From  the  principles  of  limits,  it  follow^s  that  if  two  functions 
an  continuous  at  a  given  point,  their  sum,  difference,  and  prod- 
uct are  continuous  at  that  point.  As  an  immediate  conse- 
quence, every  rational  entire  function  of  z  is  continuous  at 
ev(ry  finite  point ;  for  every  such  function  can  be  constructed 
fron  z  and  constant  quantities  by  a  finite  number  of  additions, 
sultractions,  and  multiplications. 

Let  a  function  of  a  single  independent  variable  be  contin- 
U015  at  c,  and  let  it  take  at  that  point  the  value  /,  different 
fron  zero.  Suppose  also  that  at  any  other  point  ^-|- J^  the 
function  takes  the  value  /  -j-  At.     Then 

I  I  At 


t  +  At      t  /(/  +  At) 

T  it  be  assumed  that  |  J/ 1  <  |  /  |,  the  modulus  of  the  preceding 
cfference  cannot  exceed 

\At\ 


\t\{\t\-\At\y 

ad  will,  therefore,  be  less  than  e  if 


\Ji\< 


I+6l/! 


Hence  if  a  function  is  continuous  and  different  from  zero 
t  a  point  c,  its  reciprocal  is  also  continuous  at  c.  It  follows 
t  once  that  if  two  functions  are  both  continuous  at  c,  their 
itio  is  continuous  at  c,  unless  the  denominator  reduces  to  zero 


232  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

at  that  point.  But  every  rational  function  of  2  may  be  expressed 
as  the  ratio  of  two  entire  functions.  It  is  therefore  continuons 
for  all  values  of  z  except  those  for  which  its  denominator 
vanishes. 

Consider  the  function  exp^, 

Hence  if  |^^|<i, 


but  the  limit  of  the  second  member  is  zero  when  |z/^|  ap- 
proaches zero.  Hence  exp  2  is  continuous  for  all  finite  valies 
of  JS. 


Prob.  5.  Show  that  cos  z  and  sin  z  are  continuous  for  all  mite 
values  of  z. 

Prob.  6.  Show  that  tan  z  is  continuous  in  any  circle  descibed 
about  the  origin  as  a  center  with  a  radius  less  than  -J^r.  1  \ 


Art.  6.    Graphical  Representation  of  Functions.  | 

It  was  shown  in  Art.  2  that  a  plane  suffices  for  the  complete 
graphical  representation  of  the  values  of  an  independent  va^ 
able.     In  the  same  way  it  is  convenient  to  use  a  second  plae 
to  represent  graphically  the  values  of  any  one-valued  functio. 
For  example,  if  w  =f{z)  be  such  a  function,  to  each   poit 
X  ~\-  iy  oi  the  independent  variable   will   correspond  a  poivt 
21  _|_  iij  of  the  function.    This  point  u  -\-  iv  is  called  the  "  image '"^ 
of  the  point  x  -\-iy.     If  w  is  a  continuous  function  of  z,  then 
every  continuous  curve  in  the  ^-plane  will  have  an  image  ii] 
the  w-plane,  and  this  image  will  be  also  a  continuous  curve.      I 

Consider  the  expression  u  +  iv  =  ;ir'  +  y  -f-  2ixy.      Herj 


Art.  7.] 


DERIVATIVES. 


233 


u  =  x'^  -{-  y  and  v  =  2xy,  Since  to  every  value  of  js  corre- 
spond determinate  values  of  x  and  j, 
and  consequently  determinate  values 
of  u  and  v,  this  expression  falls  un- 
der the  general  definition  of  a  func- 
tion of^.  It  is  evidently  continuous. 
Every  straight  line  x  =  t  parallel  to 
the  axis  of  j  is  converted  by  means 
of  it  into  a  parabola  v^  =  ^f{u  —  /'). 

Prob.  7.  Find  the  family  of  curves 
into  which  the  straight  lines  parallel  to 
the  axis  of  _y  are  converted  by  means  of 
the  function  u  -\-  iv  =  x^  ^  y^  -\-  2txy, 
of  this  family  intersect. 


Show  that  no  two  curves 


Art.  7.    Derivatives. 

Let  w  =  f{z)  be  a  given  function  of  2.  If  h  is  an  "  infini- 
tesimal," that  is,  a  variable  having  zero  as  its  limit,  and  if  the 
expression 

f{z  +  h)-f{z) 
h 

has  a  finite  determinate  limit,  remaining  the  same  under  all 
possible  suppositions  as  to  the  v^ay  in  which  ^approaches  zero, 
this  limit  is  said  to  be  the  "  derivative  '*  of  the  function  f{z)  at 

the  point  z.     In  this  case  w  =  f{z)  is  said  to  be  "  monogenic  " 

div 
at  z.     The  derivative  is  written  f'{z)  or  -r-.     A  function  is  said 

to  be  monogenic  in  a  region  of  the  plane  of  the  independent 
variable  if  it  is  monogenic  at  every  point  of  that  region. 

Consider  now  the  circumstances  under  which  a  function 
w  ^:^  u  -\-  iv  may  have  a  derivative  at  the  point  z  ■=^  x  A;-  iy. 
If  z  be  given  a  real  increment,  x  is  changed  into  x  -j-  Ax,  while 
y  is  unaltered,  so  that  Az  =^  Ax',  and 

Aw   ^  Au         .Av 
Az    ~  Ax  ~^    Ax' 


234  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

If,  on  the  Other  hand,  z  is  given  a  purely  imaginary  incre- 
ment, Az  =  iAy^  and 

Aw  _  Au  Av 

Az    ~  iAy        Ay  ' 

If  the  second  members  of  these  equations  approach  deter- 
minate limits  as  Ax  and  Ay  approach  zero,  and  if  these  limits 
are  equal, 

dx^'^'dx~      ^  dy      dy 
Hence,  equating  real  and  imaginary  parts, 

a^^  _  a^        ^  —  _2?f 

dx  ~~dy'         dx~      dy' 
which  are  necessary  conditions  for  the  existence  of  a  derivative. 

It  can  be  shown  that  these  conditions  are  also  sufficient. 
For  let  the  increment  of  the  independent  variable  be  entirely 
arbitrary,  no  supposition  being  made  as  to  the  relative  magni- 
tudes  of  its  real  and  imaginary  parts.  Then  the  differential  of 
the  function,  that  is,  that  part  of  the  increment  of  the  function 
which  remains  after  subtracting  the  terms  of  order  higher  than 
the  first,  is 


^  \dx  ^  dxJ      ^\dy^dyl-^ 

du  +  idv       \dx~^   a  J  "^  \dy         dy)  dx 


dx-\-idy  iJri— 

dx 

which,  by  virtue  of  the  conditions  written  above,  is  equal  to 
either  member  of  the  equation 

'dx  ~^    'dx  ^dy  "^   dy 

The  value  thus  obtained  is  independent  of  --^,  or,  what  is  the 

dx 

same  thing,  of  the  direction  of  approach  to  the  point  z.     The 


Art.  7.]  DERIVATIVES.  235 

existence  of  a  derivative  of  the  function  w  depends,  therefore, 

only  on  the  existence  of  partial  derivatives  — -,    tt-,    — ,    tt- 
^  dx     dx      dy     dy 

satisfying  the  specified  equations  of  conditions. 

The  same  equations  of  condition  express  the  fact  that 
w  =  u  -{-  iVy  supposed  to  be  an  analytical  expression  involving 
X  and  y,  involves  -sr  as  a  whole,  that  is,  may  be  constructed  from 
z  by  some  series  of  operations,  not  introducing  x  or  y  except 
in  the  combination  x  -\-  iy.  In  other  words,  they  indicate  that 
X  and  y  may  both  be  eliminated  from  w  =  <p{x,  y)  by  means  of 
the  equation  2  =  x  -{-  ty.  This  property  might  have  been  used 
to  define  monogenic  function,  but  such  a  definition  would  have 
had  the  disadvantage  of  assuming  a  priori  that  the  function 
was  capable  of  analytical  expression  in  terms  of  the  independ- 
ent variable. 

A  monogenic  function  is  necessarily  continuous  ;  that  is, 
the  existence  of  a  derivative  involves  continuity.     For,  if 

hmit  i — ^-^^  =/  {^)y 

it  follows  that 

where  7;  approaches  zero  with  k.  Hence  /{z)  is  the  limit  of 
/{z -\-  K)  when  h  approaches  zero,  or  f{z)  is  continuous  at  the 
point  -sr. 

The  following  pages  relate  almost  exclusively  to  functions 
which  are  monogenic  except  for  special  isolated  values  of  z. 
Functions  which  are  discontinuous  for  every  value  of  the  inde- 
pendent variable,  and  functions  which  are  continuous  but  admit 
no  derivatives,  have  been  little  studied  except  in  the  case  of 
real  variables.* 

*  In  this  connection  see  G.  Darboux,  Sur  les  fonctions  discontinues,  An- 
nales  de  I'^cole  Normale,  Series  2,  Vol.  4(1875),  pp.  51-112.  For  a  systematic 
treatment  of  functions  of  a  real  variable,  see  the  German  translation  of  Dini's 
treatise  by  Lttroth  and  Schepp,  Leipzig,  1892. 


^36  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [ChAP.  VI. 

Art.  8.    Conformal  Representation. 

Let  z  start  from  the  point  z^  and  trace  two  different  paths 
•forming  a  given  angle  at  the  point  z^,  and  let  z^  and  z^  be  arbi- 
trary points  on  the  first  and  second  paths  respectively.     Then 

z,  —  z^  =  rj(cos  6/j  -{-ism  6^)  =  r/'^s 

where  r^  denotes  the  length  of  the  straight  line  joining  z^  and 


z^ ,  and  Oj^  denotes  the  inclination  of  this  line  to  the  axis  of 
reals.     In  the  same  way,  for  the  point  z^ ,  there  is  an  equation 

z^  —  z^=  r, (cos  6^  +  z  sin  6^)  =  r^e'\ 

If  now  w  is  a.  one-valued  monogenic  function  of  Zy  in  the 
region  of  the  .a-plane  considered,  to  the  points  z„  z^,  z^  corre- 
spond points  w^j  w^,  w^;  and  for  these  points  can  be  formed 
the  equations  / 

9  ..Q^ 


w. 


^0  =  p/  'y     w^  —  w,=  p^e^\ 


From  the  supposition   that  w  is  monogenic,  it  follows  at 
once  that,  when  z,  and  z^  are  assumed  to  approach  z^, 

limit  ??i-^  =  limit  ^'-^0. 

If  the  members  or  this  equation  are  not  equal  to  zero,  it  may 
be  put  in  the  form 


limit 


w. 


w„ 


^-.  =  limit  ^LZlfo, 


w„ 


Art.  8.]  CONFORMAL    REPRESENTATION.  23( 


or 


Hence 


limit  ^^V^"^'^'-*'^  =  limit  "L^e'^''-'^. 


limit  (0,—  0,)  =  limit  (^,-.  6^,) ; 


and  the  images  in  the  z£/-plane  of  the  two  paths  traced  by  ^ 
form  at  w^  an  angle  equal  to  that  at  ^„  in  the  .s-plane.  Accord- 
ingly, if  2  be  supposed  to  trace  any  configuration  whatever 

in  a  portion  of  the  ^-plane  in  which  — — -  is  determinate  and  not 

a2 

equal  to  zero,  every  angle  in  the  image  traced  by  w  will  be 

equal  to  the  corresponding  angle  in  the  ^-plane.     If,  for  exam-^ 

pie,  such  a  portion  of  the  <2r-plane  be  divided  into  infinitesimal 

triangles,  the  corresponding   portion    of    the   <sr-plane  will   be 

divided  in  the  same  manner,  and  the  corresponding  triangles 

will  be  mutually  equiangular.     Such  a  copy  upon  a  plane,  or 

upon  any  surface,  of  a  configuration  in  another  surface  is  called 

a  "  conformal  representation." 

^^    . 

-;—    is  the 

'*  magnification."  Its  value,  which,  in  general,  changes  from 
point  to  point,  may  be  obtained  from  the  relations 


Idiv 
—r 
\dz 


limit 


dw 


dz 


■=(S)V(S)'=(i)+i)' 


'du  pv  _  du  dv 
'dx   dy       dy  dx 


The  theory  of  conformal  representation  has  interesting  ap- 
plications to  map  drawing.* 

*  For  the  literature  of  the  subject,  see  Forsyth,  Theory  oi  Functions,, 
p.  500,  and  Holzmuller,  Einfiihring  in  die  Theorieder  isogonalen  Verwandschaf- 
ten  und  der  conformen  Abbildungen,  verbunden  mit  Anwendungen  auf  mathe* 
niatische  Physik. 


^38  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

Art.  ...    Examples  of  Conformal  Representation. 

Case  I. — Let  w  =  ^  -j-c.  This  function  is  formed  from  the 
independent  variable  by  the  addition  of  a  constant.  Putting 
for  w,  z,  and  c,  respectively,  u  +  iv,  x -\- iy  and  a  +  ib,  one  ob- 
tains 


x-\-a, 


■y+b. 


Any  configuration  in  the  ^-plane  appears,  therefore,  in  the 
ze/-plane  unaltered  in  magnitude,  and  is  situated  with  respect  to 
the  axes  as  if  it  had  been  moved  parallel  to  the  axis  of  reals 
through  the  distance  a  a'nd  parallel  to  the  axis  of  imaginaries 
through  the  distance  b.  The  following  diagrams  represent  the 
transformation  of  a  network  of  squares  by  means  of  the  rela- 
tion w  ^  z  -\-  c. 


Case  II.  —  Let  w  =  cz.     Writing  w  =  p^*'*,   z  =  re^^,  and 
{:  =  r^e^\  the  following  equations  result  : 

p  =  r^r,     (p  z=  e^-\-  d. 

The  origin  transforms  into  the  origin,  all  distances  measured 
from  the  origin  are  multiplied  by  a  constant  quantity,  and 
all  straight  lines  passing  through  the  origin  are  turned  through 
a  constant  angle.     See  the  following  diagrams. 


Art.  9.1         examples  of  conformal  representation. 


239 


Case  III. — Let  zv  =  e".     Writing  js  =  x  -{•  zj/,  the  function 
becomes 

w  =  e^e^^  =  e''{cosy  -f  ^  sin_^). 

Every  straight  line  x  =  t^  parallel  to  the  axis  of  y  is  trans- 
formed into  a  circle  p  =  ^»  described  about  the  origin  as  a 
center,  the  axis  oi  y  becoming  the  unit  circle.  Points  to  the 
right  of  the  axis  oi  y  fall  without  the  unit  circle,  while  points 
to  the  left  of  this  axis  fall  within.  Every  straight  line  y  =  t^ 
parallel  to  the  axis  of  x  becomes  a  straight  line  v/u  =  tan  t^ 
passing  through  the  origin.  The  accompanying  diagrams* 
exhibit  in  a  simple  manner  the  periodicity  expressed  by  the 

equation 

exp  {2  -j-  2n7rt)  =  exp  (<sr), 

where  n  is  any  positive  or  negative  integer. 

To  every  point  in  the  w-plane,  excluding  the  origin,  corre- 
spond an  infinite  number  of  points  in  the  <3f-plane.  These 
points  are  all  situated  on  a  straight  line  parallel  to  the  axis  of 


*The  figures  of  this  and  the  following  example  are  taken  from  Holzmilller's 
treatise. 


240 


FUNCTIONS    OF    A    COMPLEX    VARIABLE. 


[Chap.  VI. 


y^  and  divide  it  into  segments,  each  of  length  2;r.     If  z  be  one 
of  these  points,  the  general  value  of  the  inverse  function  is 

log  w  =^  z'  -{•  2ni7t, 

where  n  is  any  positive  or  negative  integer. 

If  any  straight  line  beginning  at  the  origin  be  drawn  in  the 
tt^plane,  there  will  correspond  in  the  ^-plane  an  infinite  number 


^Tt- 


^TT 


M 


of  straight  lines  parallel  to  the  axis  of  Xy  dividing  that  plane 
into  strips  of  equal  width.  To  any  curve  in  the  zc/-plane 
which  does  not  meet  the  line  just  drawn,  will  correspond  in 
the  ^-plane  an  infinite  number  of  curves,  of  which  there  will  be 
one  in  each  strip. 

Case  IV. — Let  w  =  cos  z.  Writing  w=u-{-tv,z=x-\-  /r, 
and  employing  as  equations  of  definition  cos  (^j/)  =  cosh ;/, 
sin  (ty)  =  ^  sinhj^,  the  given  function  takes  the  form 


Hence 


u  -\-  iv  =  cos;tr  cosh  j/  —  z  sin  x  sinhj^. 

t 
u  =  cos  X  cosh  j/j  V  =  —  sin  x  sinh_;/. 


c-t^-^y/^j 


^u 


Art.  9.] 


EXAMPLES    OF    CONFORMAL    REPRESENTATION. 


241 


Any  straight   line,  x  =  /j,  parallel  to  the  axis  of  ^,  is  trans- 
formed into  one  branch  of  a  hyperbola, 


V' 


=  I, 


Y 


If 


cos"  /,      sin"  /, 
X  X 

having  its  foci  at  the  points  +  i  and  —  i.      Any  straight  line, 

y  =  t^^  parallel  to  the  axis  of  x,  is  transformed  into  an  ellipse. 


+ 


I, 


cosh'  /,   '   sinh'  /„ 

having  its  foci  at  the  same  points,  any  segment  of  the  straight 
line  equal  in  length  to  2n  corresponding  to  the  entire  curve 
taken  once.  By  means  of  these  confocal  conies,  the  w-plane 
is  divided  into  curvilinear  rectangles,  the  conformal  represen- 
tation  breaking  down  only  at  the  foci,  where  the  condition 


that  -;—  should   be  different  from  zero  is  not   fulfilled. 

az 
periodicity  of  the  function,  expressed  by  the  equation 

cos(^  -\-  27r)  z=  COSTS', 


15 


16 


H 


10 


12 


13 


M 


U 


15 


10 


The 


16 


15 


12 


11 


10 


16 


15 


is  exhibited  graphically 
in  the  accompanying 
diagrams. 

It  is  interesting  to 
note  in  this  example, 
as  also  in  the  preceding 
one,  that  the  conformal 
representation  intro- 
duces well-known  sys- 
tems of  curvilinear 
coordinates,  the  cartesian  coordinates,  ;r,  j  of  a  point  in  th^ 


\  /\  11  \ 

12 

V 

13 

/    ^*    /\     ^ 

L 

M 

\  ^  )k^  ^ 

E 
5 

D 
i 

c  \^bV  1  1 

242 


FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [C^AP.  VI. 


^-plane  serving  to  determine  its  image  in   the  w-plane  as  an 
intersection  of  orthogonal  curves. 

Case  V. — Let  w  —  z\     Writing  w  —  u  -\-  iv,  z  =  x  -\-  iy, 
the  relations 


u  —  X 


ixy\     v=sxy-y 


follow  at  once.  If  one  of  the  variables  x,  y  be  eliminated  from 
these  two  equations  by  means  of  the  equation  Ix  +  fny  -\-  n  =  Q, 
representing  a  straight  line  in  the  <s:-plane,  equations  are  ob- 
tained representing  a  unicursal  cubic  in  the  «/-plane. 

By  putting  w  —  p(cos  0  -f  ^  sin  0),  z  =  r{cos  6  -\-  i  sin  d\ 
the  relations  p  =  r\  0  =  l6,  are  obtained.  Hence  the 
circle 


2   _   /.a 


gives  the  curve 


r"  —  2ar  cos  6  ■{•  a 


p5  —  2ap^  cos  -  -\-  a^  =  ^', 


which  enwraps  three  times  the  point  corresponding  to  the 
center.  The  accompanying  figure  represents  this  transfor- 
mation, the  straight  line  feg  giving  the  curve  feg. 


To  each  point  in  the  w-plane,  excluding  the  origin,  at  which 

—  =  o  and  the  conformal  representation  is  not  maintained, 
dz 


Art.  9.]  EXAMPLES    OF    CONFORMAL    REPRESENTATION. 


243 


there  correspond  three  points  in  the  ^-plane,  having  for  their 


arguments  — , 
3 


respectively.    Any  straight  line 


drawn  from  the  origin  in  the  ze^-plane  will  have,  therefore,  three 
images  in  the  ^-plane,  viz.,  three  straight  lines  diverging  from 
the  origin,  and  dividing  the  plane  into  three  equal  regions. 
Any  continuous  curve  in  the  w-plane  not  meeting  "the  line  just 
drawn  will  be  represented  in  the  ^-plane  by  three  curves,  of 
which  one  will  be  situated  within  each  of  these  regions.  In  the 
figure  here  given  are  exhibited  the  three  conformal  represen- 
tations of  a  square  formed  in  the  w-plane  by  lines  u  =  t^y  u  = 
t^^  V  =  /^,  V  =  t^,  parallel  to  the  axes. 

If  the  relation  between  w  and  ^  be  reversed,  and  z  be 
taken  as  a  function  of  Wy  z  will  be  a  three-valued  function,  its 
values  giving  rise  to  three  branches  which  will  remain  distinct 
and  continuous  except  when  w  becomes  equal  to  zero. 

(^ 

y 


d      c 
a      b 


Prob.  8.  li  w  =  z  -\ ,  show  that  circles  in  the  2^-plane  having 

z 

a  common  center  at  the  origin  transform  into  confocal  ellipses. 

z  —  i 


Prob.  Q.  li  w  = :,  show  that  the  axis  of  reals  in  the  2r-plane 

^  z  ^  t 

I,  and  the  upper  half  of  the  -s-plane 


z  -\-  I 
transforms  into  the  circle  \w\  - 
into  the  interior  of  this  circle. 


244  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VL 

Art.  10.  CoNFORMAL  Representation  of  a  Sphere. 

Let  OPO'  be  a  sphere  having  its  diameter  00'  equal  in 

length  to  unity.  Con- 
struct tangent  planes  at 
at  O  and  O' .  Draw  in 
the  tangent  plane  at 
O  rectangular  axes  Ox 
and  Oy  ;  and  in  the 
other  plane  draw^  as 
axes  O'u,  parallel  to  Ox 
and  measured  in  the 
same  direction,  and  O'v 
parallel  to  Oy  but  meas- 
ured in  a  contrary  di- 
rection. Join  any  point 
z  in  the  plane  xOy  to 
O'  by  a  straight  line,  and  let  O' z  meet  the  sphere  in  P.  Draw 
(9Pand  produce  it  to  meet  the  plane  uO'v  in  w. 
From  the  similar  triangles  O'  Oz  and  OO'w 

Oz 


00' 


00' 

O'w 


or     Oz .  O'w  =  00'  ; 


that  is, 


w 


rp 


To  an  observer  standing  on  the  sphere  at  O'  rotation  about 
00'  from  O'u  toward  O'v  is  positive,  while  to  an  observer 
standing  on  the  sphere  at  O  such  a  rotation  is  negative. 
Hence 

/_xOz  =  —  /,uO'w,    or     6^  =  —  0. 

The  following  equation  results : 

wz  =  pr^^"(*+^)  =  I. 

The  w-  and  ^-planes  are  therefore  conformal  representa- 
tions of  one  another.  Any  configuration  in  one  plane  can  be 
formed  from  its  image  in  the  other  by  an  inversion  with  respect 


Art.  11.]  CONJUGATE    FUNCTIONS.  245 

to  the  origin  as  a  center,  combined  with  a  reflection  in  the  axis  of 
reals.  Such  a  transformation  was  termed  by  Cayley  a  "  quasi- 
inversion."  By  it  points  at  a  great  distance  from  the  origin 
in  one  plane  are  brought  near  together  in  the  immediate  neigh- 
borhood of  the  origin  in  the  other  plane. 

Since  the  line  0' Pz  makes  the  same  angle  with  the  plane 
tangent  to  the  sphere  at  P  as  with  the  plane  xOy,  any  spherical 
angle  having  its  vertex  at  P  is  projected  into  an  equal  angle  at 
z.  The  sphere  is  thus  seen  to  be  related  conformally  to  the 
plane  xOy^  and  it  must  be  also  so  related  to  the  plane  uO'v, 

The  representation  of  the  sphere  upon  a  tangent  plane  in 
the  manner  described  above  is  termed  a  "  stereographic  pro- 
jection." When  to  this  representation  is  applied  a  logarithmic 
transformation,  that  is,  one  inverse  to  the  transformation 
described  in  Case  III  of  the  preceding  article,  the  so-called 
"  Mercator's  projection"  is  obtained. 

Art.  11.    Conjugate  Functions. 

The  real  and  imaginary  parts  of  a  monogenic  function, 
w  =  u-\-  iv,  have  been  shown  to  satisfy  the  partial  differential 
equations 

9^  _  9^  dv  _  _  du 

dx  "  dy  'dx  ~       ^' 

At  any  point,  therefore,  where  u  and  v  admit  second  partial 
derivatives,  one  obtains 

that  is,  the  functions  u  and  v  are  solutions  of  Laplace's  equa- 
tion for  two  dimensions.  Any  two  real  solutions  /  and  q  of 
this  equation,  such  that  p-\-iq  is  a  monogenic  function  of 
X  -\-  iy,  are  called  ''  conjugate  functions."  *  Thus  the  examples 
of  Art.  9  furnish  the  following  pairs  of  conjugate  functions: 

*  Maxwell,  Electricity  and  Magnetism,  1873,  vol.  i,  p.  227. 


246  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

X  -{-  a,  y  -{-  d  ;  r^r  cos  {d^  +  ^)'  ^i^  sin  (6^  +  ^) ;  ^^  cos  j,  e''  sin^ ; 
cos  X  cosh  J,  —  sin  x  sinh  j  ;  x^  —  3^7^  ^xy^  —  y^-  The  second 
pair  is  expressed  in  polar  coordinates,  but  may  be  transformed 
to  cartesian  coordinates  by  means  of  the  relations 

r  =   Vx'+f,     cos  6  =  -    "^  sin  0  =  —-A 


Vx'+f  Vx'+f 

If  one  of  two  conjugate  functions  be  given,  the  other  is 
thereby  determined  except  for  an  additive  constant.  Let  Uy 
for  example,  be  given.     Then 

dv  =  — dx  -\-  —dy 

^x         -dy 

=  —  — dx -\-  — dy. 

dy     ^  dx  ^' 

and  therefore  the  value  of  v  is 

r{-^Jidx^.^dy\ 

The  equations  u  -^  c^,  v  =^  c^,  obtained  by  assigning  con- 
stant values  to  two  conjugate  functions,  represent  in  the 
w-plane  straight  lines  parallel  to  the  coordinate  axes.  It 
follows  that  the  curves  which  these  equations  define  in  the 
<8'-plane  intersect  at  right  angles.  Consequently,  by  varying 
the  quantities  c^  and  ^,,  two  orthogonal  systems  of  curves  are 
obtained  ;  and  c^  and  c^  may  be  taken  as  orthogonal  curvilinear 
coordinates  for  the  determination  of  position  in  the  ^-plane. 

Prob.  lo.  Show  that  if  /  and  q  are  conjugate  functions  of  u  and 
Vy  where  u  and  v  are  conjugate  functions  of  x  a.ndy,p  and  q  will  be 
conjugate  functions  of  x  and  y. 

Prob.  II.  Show  that  if  u  and  tj  are  conjugate  functions  of  x  and 
J,  X  a.ndy  are  conjugate  functions  of  u  and  v. 

Art.  12.    Application  to  Fluid  Motion. 

Consider  an  incompressible  fluid,  in  which  it  is  assumed 
that  every  element  can  move  only  parallel  to  the  .s-'plane,  and 
has  a  velocity  of  which  the  components  parallel  to  the  coordi- 


Art.  12.]  application    TO    FLUID    MOTION.  247 

nate  axes  are  functions  of  x  and  y  alone.  The  whole  motion 
of  the  fluid  is  known  as  soon  as  the  motion  in  the  ^s-.plane  is 
ascertained.  When  any  curve  in  the  ^-plane  is  given,  by  the 
"flux  across  the  curve  "^  will  be  meant  the  volume  of  fluid 
which  in  unit  time  crosses  the  right  cylindrical  surface  having 
the  curve  as  base  and  included  between  the  -s-plane  and  a  par- 
allel plane  at  a  unit  distance. 

The  flux  across  any  two  curves  joining  the  points  z^  and  z 
is  the  same,  provided  the  curves  enclose  a  region  covered  with 
the  moving  fluid.  For,  corresponding  to  the  enclosed  region, 
there  must  be  neither  a  gain  nor  a  loss  of  matter.  Let  z^  be 
fixed,  and  z  be  variable.  Let  ^'  denote  the  flux  across  any  curve 
z^z,  reckoned  from  left  to  right  for  an  observer  stationed  at  z^ 
and  looking  along  the  curve  toward  z.  If  /,  m  be  the  direction 
cosines  of  the  normal  (drawn  to  the  right)  at  any  point  of  the 
curve,  and  p,  q  be  the  components  parallel  to  the  axes  of  the 
velocity  of  any  moving  element,  the  value  of  ^  will  be 


0  =    /  "{Ip  +  niq)ds. 


where  the  path  of  integration  is  the  curve  joining  z^  and  z. 
The  function  ^  is  a  one-valued  function  of  z  in  any  region 
within  which  every  two  curves  joining  z^  to  z  enclose  a  region 
covered  with  the  moving  fluid. 

If  z  moves  in  such  a  manner  that  the  value  of  tp  does  not 
vary,  it  will  trace  a  curve  such  that  no  fluid  crosses  it,  i.e.,  a 
**  stream-line."  The  curves  ip  =  const,  are  all  stream-lines,  and 
tp  is  called  the  "  stream-function."  If  p  and  g  are  continuous^ 
and  if-  z  be  given  infinitesimal  increments  parallel  to  x  and  y 
respectively,  one  obtains 

a^^~^'      Vy=^' 
If  now  the  motion  of  the  fluid  be  characterized,  as  is  the 

*  Lamb's  Hydrodynamics  (1895),  p.  69. 


248 


FUNCTIONS    OF    A    COMPLEX    VARIABLE. 


[Chap.  VI. 


case  in  the  so-called  "  irrotational"  motion,*  by  the  existence 
of  a  velocity-potential  0,  so  that 

^       90  90 

^^3^'       ^-97' 
the  following  equations  result : 

90_9^      d±^  __  90 
9-^~9^'     d^~       dy' 

Hence  0  -f  ^^  is  a  monogenic  function  of  ;r  +  iy-  The  curves 
0  =  const.,  which  are  orthogonal  to  the  stream-line's,  are 
called  the  '*  equipotential  curves." 

Consider,  as  an  example,  the  motion  corresponding  to  the 
function  f  w  =  s\     The  equipotential  curves  are  given  by  the 

equations 

u  =  x^  —  ^xy^  =  const, 

the    stream-lines    by 


the  equations 


const. 


v=^xy—y 

In  the  following  fig- 
ure the  stream-Hnes 
are  the  heavy  lines, 
while  the  equipo- 
tential curves  are 
dotted. 

The   fluid    moves 
i  in  toward  the  origin, 

which  is  called  a  "  cross-point,"  from  three  directions,  and 
flows  out  again  in  three  other  directions.  At  the  cross-point 
the  fluid  is  at  a  standstill,  since  at  that  point  the  velocity,  for 
which  the  general  expression  is 


\/'g)'+gy- 


*  In  irrotational  motion  each  element  is  subject  to  translation  and  pure 
strain,  but  not  to  rotation. 

f  F.  Klein  :  Riemann's  Theory  of  Algebraic  Functions  ;  translated  by- 
Frances  Hardcastle  (1893),  p.  3. 


Art.  12.]  application  to  fluid  motion.  249 

is  equal  to  zero.  The  stream-lines  in  the  figure  represent  the 
motion  of  the  fluid  in  each  of  six  different  angles,  as  if  the  fluid 
were  confined  between  walls  perpendicular  to  the  ^-plane. 

It  is  of  importance  to  note  that  if  the  function  considered 
be  multiplied  by  /,  the  equipotential  curves  and  stream-lines 
are  interchanged,  since  the  function  (p-^tip  then  becomes 
—  tp+  t<p. 

An  example  of  particular  interest  is 

w  =  —  jj.  log . 

2-\-  a 

Let  2  —  a  =  r/'"^s  2  -{-  a  =  r,^'^»;  then 

«  =  —  yw  log  -1,     v=  —  ju{^,  —  e^). 

The  curves  u  =  const.,  v  =  const,  form  two  orthogonal  sys- 
tems of  circles,  either  of  which  may  be  regarded  as  the  stream- 
lines, the  other  constituting  the  equipotential  curves. 


The  velocities  are  everywhere,  except  at  the  points  ±  a, 
finite  and  determinate.  If  the  circles  r,/r,  =  const,  be  taken 
as  the  stream-lines,  each  of  the  points  ±  ^  is  a  "  vortex-point." 
If  the  circles  6^  ~  6^=  const,  be  taken  as  the  stream-lines,  one 


250  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

of  the  points  ±  <2  is  a  "  source,"  the  other  a  "sink."  In  the 
latter  case,  besides  the  hydrodynamical  interpretation,  a  very 
simple  electrical  illustration  is  afforded  by  attaching  the  poles 
of  a  battery  to  a  conducting  plate  of  indefinite  extent  at  two 
fixed  points  of  the  plate. 

As  another  example  may  be  taken  the  relation  w  —  cos  z. 
As  has  been  shown,  the  curves  x  =  const,  form  a  system  of 
confocal  hyperbolas,  while  the  curves  y  =  const,  form  an 
orthogonal  system  of  ellipses.  Either  system  may  be  regarded 
as  stream-lines.  In  one  case  the  motion  of  the  fluid  would  be 
such  as  would  occur  if  a  thin  wall  were  constructed  along  the 
axis  of  reals,  except  between  the  foci,  and  the  fluid  should  be 
impelled  through  the  aperture  thus  formed.  In  the  other  case 
the  fluid  would  circulate  around  a  barrier  placed  on  the  axis  of 
reals  and  included  between  the  foci. 

Besides  their  application  to  fluid-motion,  conjugate  func- 
tions have  important  applications  in  the  theory  of  electricity 
and  magnetism  "^  and  in  elasticity.f 

Art.  13.    Critical  Points. 

Let  w  be  any  rational  function  of  2.  It  can  be  written  in 
the  form 

where /(-s-)  and  0  {£)  are  entire  and  without  common  factors. 
This  function  is  finite  and  admits  an  infinite  number  of  suc- 
cessive derivatives  for  every  finite  value  of  ^,  except  llie  roots 
of  the  equation  0  (-S")  —  o.  Let  a  be  such  a  root.  Then  the 
reciprocal  of  the  given  function  is  finite  and  admits  an  infinite 
nnmber  of  successive  derivatives  at  the  point  a.     Such  a  point 

*  J.  J.  Thomson,  Recent  Researches  in  Electricity  and  Magnetism  (1893)^ 
p.  208. 

f  Love,  Theory  of  Elasticity  (1892),  vol.  i,  p.  331. 


Art.  13.]  critical  points.  25L 

is  called  a  "pole."  Any  rational  function  having  a  pole  at  a 
can  be  put  by  the  method  of  partial  fractions  in  the  form 

z  —  a  ^    \z  —  of  ^   ^^  ' 

where  ^,,  . .  .,  ^^  are  constants,  A^  being  different  from  zero,, 
and  ip{z)  is  finite  at  the  point  a.  The  integer  k  is  said  to  be 
the  "  order  "  of  the  pole,  and  the  function  is  said  to  have  for 
its  value  at  a  infinity  of  the  y^th  order.  In  accordance  with 
the  definition  of  a  derivative,  w  does  not  admit  a  derivative  at 
a.  From  the  character  of  the  derivative  in  the  immediate 
neighborhood  of  a,  however,  the  derivative  is  sometimes  said 
to  become  infinite  at  a. 

The  trigonometric  function  cot<sr  has  a  pole  of  the  first 
order  at  every  point  z  =  mn,  m  being  zero  or  any  integer  posi- 
tive or  negative. 

The  function  w  =  log  (z  —  a)  has  for  every  finite  value  of 
z,  except  ^  =  ^,  an  infinite  number  of  values.  If  ^  —  ^  is  writ- 
ten in  the  form  7?/®, 

where  log  R  is  real,  and  m  is  zero  or  any  positive  or  negative 
integer.  If  z  describes  a  straight  Hne,  beginning  at  a,  0  will 
remain  fixed,  but  R  will  vary.  The  images  in  the  ^-plane  will 
therefore  be  straight  lines  parallel  to  the  axis  of  reals,  dividing 
the  plane  into  horizontal  strips  of  width  2;r.  If  now  the  ^-plane 
is  supposed  to  be  divided  along  the  straight  line  just  drawn, 
and  z  varies  along  any  continuous  path,  subject  only  to  the 
restriction  that  it  cannot  cross  this  line  of  division,  there  will 
be  a  continuous  curve  as  the  image  of  the  path  of  z  in  each 
strip  of  the  ^e'-plane.  Each  of  these  images  is  said  to  corre- 
spond to  a  ''branch"  of  the  function,  or,  expressed  otherwise, 
the  function  is  said  to  have  a  branch  situated  in  each  strip. 
The  line  of  division  in  the  -s'-plane,  which  serves  to  separate 
the  branches  from  one  another  is  called  a  "  cut." 


252  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

At  the  point  ^  =  a  no  definite  value  is  attached  to  the 
function.  As  z  approaches  that  point  the  modulus  of  the  real 
part  of  the  function  increases  without  limit,  while  the  imagi- 
nary part  is  entirely  indeterminate. 

Let  js^  be  an  arbitrary  point,  distinct  from  a,  and  let 

log  R^  -\-  iS^  -\-  2m7ci 

be  any  one  of  the  corresponding  values  of  the  function.  Sup- 
pose that  z  starts  from  2^  and  describes  a  closed  path  around 
the  point  a,  the  values  of  the  function  being  taken  so  as  to 
give  a  continuous  variation.  Upon  returning  to  the  point  2^ 
the  value  of  the  function  will  be 

log  R,  +  iO^  +  2{m  +  i)7ci, 
or  log  R,  +  /©„  +  2(m  -  i)7tt, 

according  as  the  curve  is  described  in  a  positive  or  negative 
direction.  By  repeating  the  curve  a  sufficient  number  of  times 
it  is  evidently  possible  to  pass  from  any  value  of  the  function 
at  2^  to  any  other.  When  a  point  is  such  that  a  <sr-path  en- 
closing it  may  lead  in  this  manner  from  one  value  of  a  function 
to  another  value,  it  is  called  a  "  branch-point."  In  the  case 
of  the  function  here  considered,  the  point  ^  =  ^  is  called 
a  "logarithmic  branch-point,"  or  a  point  of  "logarithmic 
discontinuity." 

The  function  w  =  log  ^^-^,  where  /(2)  and  (^(2)  are  entire, 

<p{2) 

has  a  point  of  logarithmic  discontinuity  at  every  point  where 
either /(^)  or  0(^)  is  equal  to  zero.     For,  writing 

M  =  A{2  -  ayi2  -  a,y^ ... 

(p(2)  =  B{2  -  d,y^{2  —  d^y^ . . . 

the  value  of  w  may  be  written 

w  =  log  -  +  :^p^  log  (2  -  a^)  -  2g„  log  (2  -  b„), 

JD  rn  n 


Art    13]  CRITICAL   POINTS.  253 

Take  now  the  function  w  =  e^.  It  has  a  single  finite  value 
for  every  value  of  2  except  2  =  0.  If  -sf  is  supposed  to  ap- 
proach zero,  the  limit  of  the  value  of  the  function  is  indeter- 
minate. 

For  let  /  +  tg  be  perfectly  arbitrary,  and  write 

If  now  a  +  zd  is  the  reciprocal  of  /  -|-  ^^>  so  that 
a  =      ^  ^  —         ^ 


the  preceding  equation  may  be  written 

1 
ga^ib  ==  r  -|-  id. 

But  whatever  the  value  of  the  integer  m,  q-\-  2mn  may  be 
substituted  for  q  without  altering  the  value  of  ^  +  ^<^»  ^^^  hence 
both  a  and  b  may  be  made  less  than  any  assignable  quantity. 

The  given  function  e'»  therefore  takes  the  value  c  +  id  at  points 
a  +  ib  indefinitely  near  to  the  origin.  A  point  such  that,  when 
z  approaches  it,  a  function  elsewhere  one-valued  tends  toward 
an  indeterminate  limiting  value  is  called  an  "  essential  sin- 
gularity." 

1 
Prob.  12.  Show  that  for  the  function  e'-'^  z  =  a  is  an  essential 

singularity. 

_i_ 
Prob.  13.  The  function  €  ^*  considered  as  a  function  of  a  real 
variable  is  continuous  for  every  finite  value  of  z,  and  the  same  is 
true  of  each  of  its  successive  derivatives.  Show  that  when  it  is 
regarded  as  a  function  of  a  complex  variable,  s  =  o  is  an  essential 
singularity. 

In  order  to  illustrate  still  another  class  of  special  points 
take  the  function 


w  =  V(z  —  aX^  —  a,)  .  .  .{z  —  a„). 


264: 


FUNCTIONS    OF    A    COMPLEX    VARIABLE,  [ChaP.  VI. 


This  function  has  at  every  finite  point,  except  a^,  a^,  .  . . ,  a„y 
.two  distinct  values  differing  in  sign.  At  these  points,  however, 
it  takes  but  a  single  value,  zero.  From  each  of  the  points 
a^^  a^,  .  .  . ,  a„  let  a  straight  line  of  indefinite  extent  be  drawn  in 
such  a  manner  that  no  one  of  them  intersects  any  other,  and 
•suppose  the  ^-plane  to  be  divided,  or  cut,  along  each  of  these 
lines.  Along  any  continuous  path  in  the  -s-'plane  thus  divided 
the  values  of  the  function  form  two  distinct  branches. 

For,  writing 


js  —  a,  =^  r,e^ 


z  —  a„ 


r/^'^y 


r^e' 


the  function  takes  the  form 


e.  +  fla-f 


No  "closed  path  in  the  divided  plane  will  enclose  any  of  the 


points  a^ ,  a^ 


and  the  quantities  ^i ,  6^,,  .  .  .  ,  ^«,  after 


<:ontinuous  variation  along  such  a  path,  must  resume  at  the 
initial  point  their  original  values.  No  such  path,  therefore,  can 
lead   from   one  value  of  the  function  at  any  point  to  a  new 

value  of  the  function  at  the 
same  point.  If,  however,  the 
cuts  are  disregarded  and  z 
traces  in  a  positive  direction, 
a  closed  curve  including  an  odd 
number  of  the  points  <2, ,  ^,, 
.  .  .  ,  <2:„,   and    not    intersecting 

itself,  then   an   odd  number  of 

the  quantities  6^, ,  6^,,  .  .  .  ,  ^„  are  each  increased  by  2;r;  and 
the  value  of  the  function  is  altered  by  a  factor  ^(S^+D'^*^  and 
so  changed  in  sign.  In  the  same  way  any  closed  path  de- 
scribed about  one  of  these  points,  and  enwrapping  it  an  odd 
number  of  times,  leads  from  one  value  of  the  function  to 
the  other.  On  the  other  hand,  a  simple  closed  path  enclosing 
an  even  number  of  these  points,  or  a  closed  path  which  en- 
closes but  one  of  the  points,  enwrapping  it  an  even  number  of 
times,  leads  back  to  the  initial  value  of  the  function.     It  fol- 


Art.  13.]  CRITICAL    POINTS.  255 

lows  that  each  of  the  points  a^ ,  a^,  ,  .  .  ,  a„is  a.  branch-point. 
Any  point  in  the  ^-plane,  closed  paths  about  which  lead  from 
one  to  another  of  a  set  of  different  values  of  a  function,  the 
number  of  values  in  the  set  being  finite,  is  called  an  ''algebraic 
branch-point." 

As  a  further  illustration,  consider  the  function 
w  =  2^  -\-(2  —  a)^, 
which  is  a  root  of  the  equation  of  the  sixth  degree, 

w'  —^2zv*  —  2{z  —  d)w^  +  ^z^w^  —  6z{z  —  d)w  -\-  {z—df—z^^=o. 

The  function  has  at  every  point,  except  z  =  o  and  z  =^  a, 
six  distinct  values.  Six  branches  are  thereby  formed  which 
can  be  completely  separated  from  one  another  by  making  cuts 
from  the  points  z  =^o  and  ^  =  ^  to  infinity.  Putting  od  for  the 
cube  root  of  unity,  these  six  branches  can  be  written 

1/2      ,  ,  sl/3  1/2  .  1/3 

w^  =  z  -\-  {z  —  a)'  ,  w^—  —  z  +  {z  —  a)'  ^ 
w^  —  z^^  -\-  Go{z  —  af^\  w^=±  —  z^  -\-  oo{z  —  d)^^ , 
w^  =  z^^  +  oo'iz  —  dj^^,     Wg  =  —  z^^  +  gd''{z  —  af^^. 

The  branches  w^  and  w^,  w^  and  w^,  w^  and  w^  are  interchanged 
by  a  small  closed  circuit  described  about  s-^o,  while  a  small 
circuit  described  about  z=^  a  permutes  cyclically  the  branches 
w^,  W3,  W5,  and  also  the  branches  «/,,  w^,  w^. 

All  of  the  special  points  examined  above,  poles,  points  of 
logarithmic  discontinuity,  essential  singularities,  and  branch- 
points, are  called  critical  points.  In  fact,  a  function,  or  a 
branch  of  a  function,  is  said  to  have  a  "  critical  point "  at  each 
point  where  it  fails  to  have  a  continuous  derivative,"^  or  about 
which  as  a  center  it  is  impossible  to  describe  a  circle  of  deter- 
minate radius  within  which  the  function,  or  branch,  is  one- 
valued. 

Any  point  not  a  critical  point  is  called  an  "■  ordinary  point." 

*  Continuity  and,  therefore,  finiteness  of  the  function  are  implied  in  the 
existence  of  a  derivative. 


256  FUNCTIONS   OF   A   COMPLEX   VARIABLE.  [CHAP-  VL 

An  ordinary  point  at  which  a  function  reduces  to  zero  is  called 
a  "  zero  "  of  the  function. 

If  in  a  certain  region  of  the  .S'-plane  there  are  no  critical 
points  for  a  given  function,  the  function  is  said  to  be  "  synec- 
tic"  or  "  holomorphic  "  in  that  region.  If  in  a  certain  region 
the  only  critical  points  are  poles,  the  function  is  said  to  be 
"  meromorphic  "  in  that  region.  Under  similar  conditions  a 
branch  of  a  function  is  also  described  as  holomorphic  or 
meromorphic. 

Prob.  14.  When  w  and  z  are  connected  by  the  relation  w  —  g  =^ 
(z  —  /lY  show  that  if  z  describes  a  circle  about  y^  as  a  center,  m 
describes  a  circle  about  ^  as  a  center,  an  angle  in  the  2:-plane  hav- 
ing its  vertex  at  /i  is  transformed  into  an  angle  in  the  w-plane  / 
times  as  great  and  having  its  vertex  at  g,  and  that  z  =  /lis  a.  branch- 
point of  w  except  when  ^  is  an  integer. 

Art.  14.     Point  at  Infinity. 

In  determining  the  limiting  value  of  a  function  when  the 
modulus  of  the  independent  variable  2  is  increased  indefinitely, 
it  is  usual  to  introduce  a  new  independent  variable  2'  by  the 
relation  ^  =  1/2',  and  consider  the  function  at  the  point  J3'  =  o. 
This  is  equivalent  to  passing  from  the  <3'-plane  to  another  plane, 
the<8^'-plane,  related  to  the  former  by  the  geometrical  construc- 
tion described  in  Art.  10.  It  is  often  very  convenient,  however, 
to  go  further  and  to  substitute  for  the  ^-plane  the  surface  of  the 
sphere  of  unit  diameter  touching  the  ^-plane  at  the  origin.  No 
difficulty  is  thus  introduced  since,  as  explained  in  the  article 
just  cited,  any  configuration  in  the  ^-plane  obtains  a  conformal 
representation  upon  the  sphere;  and  the  advantage  is  gained 
that  the  entire  surface  upon  which  the  variation  of  the  inde- 
pendent variable  is  studied  is  of  finite  extent.  The  point  of 
the  sphere  diametrically  opposite  to  its  point  of  contact  with 
the  ^-plane  coincides  with  the  point  written  above  as  ^'  =  o. 
It  is  called  the  point  at  infinity,  ,0  =  00 ,  since  a  point  on  the 
sphere  approaches  it  at  the  same  time  that  its  image  in  the 
^-plane  recedes  indefinitely  from  the  origin. 


Art.  15.]  INTEGRAL    OF    A    FUNCTION.  257 

The  point  at  infinity  may  be  either  an  ordinary  or  a  critical 
point.  For  the  function  ^,  for  example,  it  is  an  ordinary 
point,  since  f^  =  e"' .  For  a  rational  entire  function  of  the  ;/th 
degree  it  is  a  pole  of  order  n.  Consider  it  for  the  function 
^{z  —  a^{z  —  a^ . .  .  {z  —  a^),  discussed  in  the  preceding  article. 
Let  a  circle  of  great  radius  be  described  in  the  ^-plane  inclosing 
all  the  branch-points  a^,  a,  . , .  ,  a„.  Its  conformal  representa- 
tion on  the  sphere  will  be  a  small  closed  curve  surrounding  the 
point  2  =  oo .  This  point  must,  therefore,  be  regarded  as  a 
branch-point  or  not,  according  as  the  function  changes  value  or 
not  when  the  curve  surrounding  it  is  described,  that  is  accord- 
ing as  n,  the  number  of  finite  branch-points,  is  odd  or  even* 
When  the  point  at  infinity  is  taken  into  account,  then,  the 
total  number  of  branch-points  of  this  function  is  always  even. 
The  character  of  the  point  2  =  c6  for  this  function  can  be  de- 
termined directly,  by  changing  2  into  i/z'  and  considering  the 
point  2'  =:  o. 

Prob.  15.  Show  that  2;  =  00  is  an  ordinary  point  for  -77^,  where 

cp(z)  and  ^p{z)  are  rational  and  entire  if  the  degree  of  (p{z)  does 
not  exceed  that  of  tp{z). 

Art.  15.    Integral  of  a  Function. 

Let  w  =/(2)  be  a  continuous  function  of  a  complex  vari- 
able 2y  and  suppose  2  to  describe  a  continuous  path  L  from 
the  point  2^  to  the  point  Z.  Let  a  series  of  points  2,,  2^,  ...  ,2^ 
be  taken  on  L,  and  let  /„,  /j, . . . ,  t„  be  points  arbitrarily  chosen 
on  the  arcs  2^2^,  2,2^, . . . ,  2„Z  respectively.     Form  the  sum 

s={^.-  ^„)/(/o)  +  (^,  -  ^.y(/.)  + . . .  +  (z  -  2„)/(t„). 

If  now  the  number  of  points  2^,  . .  .■,  2^  be  increased  indefi- 
nitely in  such  a  manner  that  the  length*  of  each  of  the  arcs 

*  It  is  assumed  in  regard  to  every  path  of  integration  that  the  idea  of  length 
may  be  associated  with  the  portion  of  it  included  between  any  two  of  its  points, 
or,  what  is  the  same  thing,  that  the  path  is  rectifiable.  This  condition  is  evi- 
dently satisfied  if  the  current  coordinates  x  andjj/  can  be  expressed  in  terms  of 


258 

z  z  ,  z  z  . 

0    1 '     1    a ' 


FUNCTIONS    OF    A    COMPLEX    VARIABLE. 


[Chap.  VI, 


,  ZnZ  approaches  zero  as  a  limit,  the  sum  5  ap- 
proaches a  finite  limit  which  is  inde- 
pendent of  the  choice  of  the  points  ^,, 
z^,  .,.,  z„  and  /„,  t^,...,  t„. 

For  take  any  other  sum 

formed  in  a  similar  manner.  Suppose 
—  for  the  sake  of  greater  definiteness 
that  the  points  ^^ , . . .  and  ^/, . . .  follow  one  another  on  the 
line  L  in  the  order 

,S,,  -Sj  ,  <0j  ,  Z^y  z^,  z^ , . , . , 

and  form  a  third  sum 

in  which  b  )th  series  of  points  occur.  It  may  be  shown  that  as 
the  number  of  points  in  each  of  the  series  ^,,  . . .  and  z/, ...  is 
increased,  the  differences  S'^  —  5  and  S'^  —  S'  both  approach 
zero,  from  which  it  follows  that  the  difference  S  —  S'  has  a 
limit  equal  to  zero.  For  example,  the  difference  S'^  —  S  has 
the  value 

(^,  -  ^.)[/(r.)  -AQ]  +  (^Z  -  ^,)[/(r,)  -/(A)] 

+  (^,'-<)l/{r,)-M)l  +  --. 
If  M  denotes  the  upper  extreme  of  the  quantities 

\Ar,)-M)\,         \Ar,)-M)\,         \Ard-M)\ 

the  modulus  of  S^'  —  S  will  be  less  than 

dx  ,  dy 

any  parameter  t  so  that  —  and    —  are   continuous.      For  then  the  integral 

/    sUx"^  +  dy^  is  finite.     See,  in  this  connection,  Jordan,  Cours  d'Analyse,  2d 
Edition,  Vol.  I.,  p.  loo. 


Art.  15.]  INTEGRAL    OF    A    FUNCTION.  259 

But  l-s^,  —  ^ol  is  equal  to  the  chord  of  the  arc  s^z^^  and  must 
therefore  be  less  than  or  equal  to  this  arc,  and  a  similar  result 
holds  for  each  of  the  quantities  |^/— ^i|,|^/— ^/|,  ..  •     Hence 

\S"-S\<Ml, 

where  /  denotes  the  length  of  the  path  of  integration.  When 
the  number  of  points  of  division  on  the  line  L  is  increased,  the 
differences 

/i^)  -  /('.).      /('-.)  -  /('.).      /(n)  -  At,), . . . 

decrease  indefinitely,  ior  f\s)  is  continuous.  M  acccordingly 
decreases  indefinitely  and  the  difference  S"  —  »S  approaches 
zero. 

The  limit,  the  existence  of  which  has  just  been  demon- 
strated, is  called  the  integral  of  f\^2)  along  the  path  L.     It  is 

written    j  f{z)dz.     The  definition  here  given  is  similar  to  that 

given  for  the  integral  of  a  function  of  a  real  variable.  It  is 
unnecessary  to  specify  the  path  of  integration  when  the  inde- 
pendent variable  is  restricted  to  real  values,  since  in  that  case 
it  must  be  the  portion  of  the  axis  of  reals  included  between 
the  limits  of  integration. 

The  following  well-known  principles,  applicable  to  the  case 
of  a  real  independent  variable,  may  be  readily  extended  to  the 
general  case : 

1.  The  modulus  of  the  integral  cannot  exceed  the  length  of 
the  path  of  integration  multiplied  by  the  upper  extreme  of  the 
modulus  of  the  function  along  that  path. 

2.  The  independent  variable  may  be  altered  by  any  equa- 
tion of  transformation,  but  L\  the  path  of  integration  in  the 
transformed  integral,  must  be  such  that  it  is  described  by  the 
new  variable  while  z  describes  L. 

3.  If  F{z)  is  any  one-valued  function  having  everywhere 
f{z)  for  its  derivative,  the  equation 


/ 


J{z)dz  =  F{Z)-F{z,) 


must  he  true. 


FUNCTIONS   OF    A    COMPLEX    VARIABLE.  [ChAP.  VI, 

To  prove  the  third  principle,  write  F{Z)  —  F(z^  in  the 
form 

F{Z)-F{z:)-\-F{z:)-F{3^.,)-^ ,  .  .  +F{z:)-F{z:)-\-F{z,)-F{z,),. 

Since  the  derivative  of  F{z)  is/(^), 

F{z^^^)  -  F{z^)  =  U{^m)  +  ^J(^^+,  -  2^), 

where  ?/^  has  zero  for  its  Hmit  when  z^^^  is  made  to  approach 
z^.     Hence 

F{Z)  —  F(z,)  =  limit  2/{z^){z^+,  —  ^  J  +  limit  2rf^{z„,+,  -  z,,) ; 

or,  since  the  second  term  of  the  right-hand  member  is  equal 
to  zero. 

If  no  function  F{z)  fulfilling  the  preceding  conditions  is 
known,  the  value  of  the  integral  requires  further  investi-^ 
gation. 

dz 
? 

point  .s"  =  —  I  to  the  point  z  =  i,  the  path  of  integration  being 
the  upper  half  of  the  circumference  of  a  unit  circle  described 
about  the  origin  as  a  center.  Writing  z  =  exp  (td)^  z  will 
describe  the  required  path  while  0  varies  from  7t  to  o. 

The  equations  -,  =  ^"^*^,        dz  =  ie'^dd, 
z 

dz 

—  =  ie-'^dd  =  i  cos  d  dd-\-  sin  6  dd  =  id  (sin  d)  -  d  (cos  6), 

z 

follow  at  once.     Hence  for  the  path  specified 

^rdz       .   %  ' 

J  —  =  td  (sin  6)—       d  (cos  (9)  =  —  2. 

The  application  of  the  direct  and  more  familiar  method 
gives  the  same  result : 

J   z""  ~  L      ^J^=,      L"  ^  J,=_i~ 


pdz 
Consider  as  an  example  the  integral    /  — ^  taken  from  the 


Art.  16.]    reduction  of  complex  integrals  to  real.  361 

For  a  path  along  the  axis  of   reals  between  the  limits  of 

integration  this  result  is  unintelligible.     The  discontinuity  of 

dz 
the  differential,  —^,  at  the  point  ^  =  o,  prevents  the  considera- 

z 

tion  of  such  a  path  ;  and  that  the  result  should  be  negative 
when  the  differential  is  at  every  point  of  the  path  positive 
has  no  significance.  The  introduction  of  the  complex  variable 
furnishes  a  perfectly  satisfactory  explanation  of  the  result. 

dz 
Prob.  i6.  Show  that  the  integral  of  —  along  any  semi-circum- 
ference described  about  the  origin  as  a  center  is  equal  to  ni. 


Art.  16.    Reduction  of  Complex  Integrals  to  Real. 
The  integral 

may  be  written  in  the  form 

J^{u  +  iv)(dx  +  idy\ 
or,  separating  the  real  and  imaginary  terms, 

/  {udx  —  vdy)  -\-  i     {^dx  +  udy). 

Hence  the  calculation  of  the  integral  may  be  reduced  to 
the  calculation  of  two  real  curvilinear  integrals. 

The  equations 

?^  —  ?1^    a?^  _  _  a^ 

dx  ~  dy"     ?)y'~       di 

which  express  the  condition  that  u-{-  ^V  should  be  monogenic, 
express  also  that 

udx  —  vdy,     vdx  +  tidy 

are  the  exact  differentials  of  two  real  functions  of  the  variables 
x,y.     Consider  the  case  where  these  functions  are  one-valued. 


262  FUNCTIONS   OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

Denoting  them  by  P(Xf  y)  and  Q{x,  y)  respectively,  the  inte- 
gral may  be  written 

[P{X,  Y)  -  I\x„y,)-]  +  i[Q{X,  Y)  -  Q{x,,  y,)l 

{^0,  yo)  and  {X,  Y)  being   the  initial  and  terminal  points  re- 
spectively of  the  path  of  integration. 

Art.  17.    Cauchy's  Theorem. 

Cauchy's  Theorem  furnishes  the  necessary  and  sufficient 
conditions  that  a  one-valued  function  /{s),  having  a  continuous 
derivative /'(^),  should  yield  a  one-valued  integral,  that  is,  an 
integral  the  value  of  which,  when  the  lower  limit  is  fixed,  de- 
pends simply  on  the  upper  limit,  and  not  on  the  path  of 
integration.  It  will  be  more  convenient,  before  considering 
Cauchy's  Theorem,  to  demonstrate  the  following  lemma  : 

Lemma. — l^etA  be  a  portion  of  the  ^-plane,  having  a  bound- 
ary 5  which  consists  of  a  closed  curve  not  intersecting  itself, 
or  of  several  closed  curves  not  intersecting  themselves  or  one 
another.  Denote  by  X  the  inclination  to  the  axis  of  x  of  the 
exterior  normal  at  any  point  of  the  boundary,*  that  is,  the 
normal  drawn  to  the  right  as  the  boundary  is  described  in  a 
positive  direction.  If  at  every  point  of  the  region  A,  including 
its  boundary  S,  a  function  W  of  the  real  variables  x  and  y  is 
one-valued  and  continuous   and  has  continuous  partial  deriva- 

tives ,  — ,  the  relations 

exist,  the  integrals  in  the  first  members  being  taken  along  the 

*  It  ia  assumed  that  the  boundary  has  a  determinate  tangent  at  every  point. 
If  the  boundary  of  a  given  region  is  not  of  this  sort,  the  theorem  holds  for  any 
interior  curve  of  which  this  assumption  is  true. 


Art.  17.] 


CAUCHY  S    THEOREM. 


263 


boundary  in  the  positive  direction,  and  those  in  the  second 
members  being  taken  over  the  enclosed  area. 

If  any  straight  line  parallel  to  the  axis  of  x  be  traced  in 
the  direction  of  increasing  values  of  x,  at  each  point  where 
it  passes  into  the  area  A^ 
cos  A  is  negative,  and  there- 
fore in  the  first  member  of 
{\)  dy=i  cos  \  ds  is  negative. 
At  each  point  where  this 
straight  line  passes  out  of 
the  area  A,  cos  A,  and  there- 
fore dy^  in  the  first  member 
of  equation  (i),  is  positive. 
Hence  in  the  first  member 
of  equation  (i)  the  differ- 
entials Wdy  corresponding 
to  a  given  value  of  y^  and  taken  in  the  order  of  increasing 
values  of  x,  have  signs  which,  compared  with  those  of  the 
corresponding  values  of  W,  first  differ,  then  agree,  and  so 
on  alternately.  In  order  now  to  compare  the  integral  in  the 
first  member  of  equation  (i)  with  the  integral  in  the  second 
member,  it  is  necessary  to  take  dy  as  essentially  positive. 
The  sum  of  the  differentials  in  the  first  member,  correspond- 
ing to  a  fixed  value  of  jj/,  must  therefore  be  written  in  the 
form 

dy{-W,-\-W,-W,-\-W,--,.,\ 

where  W^,  W,,...  are  the  corresponding  values  of  PF taken  in 
the  order  of  increasing  values  of  x.  But  performing  now  in 
the  second  member  of  equation  (i)  an  integration  with  respect 
to  X,  the  same  result  is  obtained,  so  that  the  two  members  of 
equation  (i)  become  identical,  and  the  equation  is  verified. 

To  obtain  equation  (2)  the  same  method  is  used.  It  is 
necessary  in  this  case  to  observe  that  if  a  line  parallel  to  the 
axis  oi  y  is  traced  in  the  direction  of  increasing  values  of  y,  at 
each  point  where  it  enters  A,  dx  in  the  integral  of  the  first 


264  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

member  must  be  taken  as  positive;  and  at  each  point  where 
this  Hne  passes  out  o(  A,  dx  in  that  integral  must  be  taken  as 
negative. 

By  means  of  the  preceding  lemma,  Cauchy's  Theorem  is 
easily  proved.     This  theorem  may  be  stated  as  follows : 

Theorem. — If,  on  the  boundary  of  and  within  a  given  region 
Ay  a  one-valued  function  w  =z  f(z)  is  monogenic,  and  its  deriv- 
ative f'{z)  is  continuous,*  the  integral  j  f{z)dz  taken  along 
the  boundary  S  is  equal  to  zero. 

For  writing  the  integral  in  the  form 

/  wdz  =  I  {udx  —  vdy)  ~\-  i  Judy  -\-  vdx), 
the  preceding  lemma  gives 

but  since  at  every  point  of  A 

the  given  integral  reduces  to  zero. 

Art.  18.    Application  of  Cauchy's  Theorem. 

From  Cauchy's  Theorem  it  follows  that,  if  two  different 
paths  Zj  and  L^  lead  from  the  point  z^  to  the  point  Z,  and  if 
along  these  paths  and  in  the  region  inclosed  between  them  a 
given  function  f{z)  has  no  critical  points,  the  integrals  of  the 
function  taken  along  these  two  paths  are  equal.  For  two  such 
paths  taken  together,  one  described  directly,  the  other  re- 
versed, constitute  a  closed  curve,  and  the  integral  taken  along 

*Otherwise  expressed,  the  one-valued  function  y^^)  has  no  critical  points  on 
the  boundary  of  or  within  A,  or /(z)  is  holomorphic  in  A. 


Art.  18.]  application  of  cauch\'s  theorem.  265 

it  is  equal  to  zero.  But,  since  reversing  the  direction  of  the 
path  of  integration  is  equivalent  to  changing  the  sign  of  the 
integral,  the  equation 

is  obtained. 

The  result  just  established  may  be  stated  in  the  following 
theorem : 

Theorem  I. — If  a  function  is  holomorphic  in  any  simply 
connected  region  bounded  by  a  continuous  closed  curve,  the 
integral  of  the  function,  from  a  fixed  lower  limit  in  that  region 
to  any  point  contained  therein,  is  independent  of  the  path  of 
integration,  and  is  a  one-valued  function  of  its  upper  Hmit. 

A  region  whose  boundary  is  composed  of  disconnected 
curves  is  not  necessarily  characterized  by  the  property  stated 
in  the  theorem.     Take,  for  example,  the  function 


and  suppose  that  o  <  |  ^,  |  <  1^,  |  <  .  .  .  <  |  <2:„  |.  With  the  ori- 
gin as  a  center,  construct  a  system  of  concentric  circles  (7,, 
6'a,  .  .  .,  Cny  C^  passing  through  ^,,  (7,  through  «„  and  so  on. 
Denote  by  5^  the  region  inclosed  within  the  first  circle  6',,  by 
S^  that  inclosed  between  6",  and  C,,  and  so  on,  the  portion 
of  the  plane  exterior  to  the  last  circle  C^  being  denoted  by  5„. 
At  an  initial  point  z^  interior  to  one  of  these  regions,  assign  to 
w  one  of  the  two  values  possible,  and  consider  the  branch  of 
w  resulting  from  a  continuous  variation.  Then  however  z  may 
vary  within  any  such  region,  this  branch  of  w  will  be  a  mono- 
genic function,  and  its  derivative  will  be  continuous.  Having 
regard  to  the  branch-points  ^,,  <2:,,  .  .  .,  a^^  it  is  evident  that  in 
the  regions  S^^  5,,  .  .  .  it  will  be  one-valued,  and  in  the  regions 
5,,  6^,.  .  .  .,  it  will  be  two-valued.  Thus  in  the  former  regions 
5o,  5,,  .  .  .,  the  branch  fulfils  all  the  conditions  required  by  the 
theorem  above.  The  theorem  is  applicable,  however,  only  to 
S^,  for  in  any  other  region  two  paths  may  be  drawn  joining 
the  same  two  points,  and  such  that  the  branch  is  not  one-valued 
throughout  the  enclosed  portion  of  the  ^-plane. 


266 


FUNCTIONS    OF    A    COMPLEX    VARIABLE. 


[Chap.  VI. 


Theorem  II. — l^/i^)  is  holomorphic  in  any  simply  connected 
region  5   bounded  by  a  continuous  closed  curve,  the  integral 

I  f{z)dz,  taken  from  a  fixed  lower  limit  z^  in  that  region  to  any 

point  Z  contained  therein,  is  a  holomorphic    function  of   its 
upper  limit. 

Let  L  be  any  path  from  z^  to  Z.  When  the  upper  limit  is 
at  the  point  Z  +  dZ,  L  followed  by  a  straight  line  from  Z  to 
Z-f-  dZ  can  be  taken  as  the  path  of  integration.     Hence 

^      ■    Az^dz--   L.Mdz=   l_ 


nz+dz  nz  nz+dz 


r*z+dz  nz+dz 

=AZ)X     dz+J^     iA^)-AZ)Yz. 


The  first  term  is  equal  to  f{Z)dZ.  The  modulus  of  second 
term  is  equal  to  or  less  than  M\  dZ\,  where  M  \s  the  upper  ex- 
treme of  |/(^)  —  f{Z)\  along  the  line  joining  Z  to  Z  -\-  aZ. 
But  since  f{z)  is  continuous,  the  limit  of  J/ when  Z -\- dZ 
approaches  Z  is  zero.     Hence 

£yAz)dz  -  £j{z)dz  =  iAZ)  +  V-\dZ, 

where  7  approaches  zero  with  dZ.     The  integral  therefore  has 
/(-Z')fora  derivative,  and  is  holomorphic  in  S. 

In  the  case  of  a  region  bounded  by  several  disconnected 
closed  curves,  of  which  one  is  exterior  to  all  the  others,, 
Cauchy's  Theorem  may  be  stated  in  the  following  form : 

Theorem  III. — Let  a  function  f\z)  be  holomorphic  in  a 
region  A  bounded  by  a  closed  curve  C  and  one  or  more  closed 
curves  C^,  C^,  .  .  .  interior  to  C.  The  integral  of  f{z)  taken 
along  C  will  be  equal  to  the  sum  of  its 
integrals  taken  in  the  same  direction 
along  the  curves  C^,  C.,,  .  .  . 

For  the   integral  of  fiz)  taken  in  a 
positive  direction  completely  around  the 
boundary  of  A    is   equal   to  zero.     But 
tne  curves  (7,,  61,,  .  .  .  are  then  described  in  the  direction  oppo^ 


Art.  19.]  theorems  on  curvilinear  integrals.  267 

site  to  that  in  which  C  is  described.     Hence  if  all  the  curves 
are  described  in  the  same  direction,  the  result  may  be  vi^ritten 


J^A^)dz=fj{z)dz  ^£f(,)d,+ 


If  there  is  but  one  interior  curve,  so  that  the  region  A  is 
included  between  two  curves  C  and  6^,,  the  integral  taken  along 
every  closed  curve  containing  (7,  but  interior  to  C  has  the 
same  value,  viz.,  the  common  value  corresponding  to  the  paths 
C  and  C^. 

Art.  19.    Theorems  on  Curvilinear  Integrals. 

Theorem  I. — If /(^)  be  continuous  in  a  given  region  except 

at  the  point  a,  the  integral  I  f{2)dzy  taken  around  a  small  circle 

*:,  having  its  center  at  «,  will  approach  zero  as  a  limit  simulta- 
neously with  the  radius  r  of  the  circle  c,  provided  only 

lim  (z  —  a)/{z)  =  o     when     z  =^  a. 

For  let  the  upper  extreme  of  the  modulus  of  {z  —  d)f{z)  on 
the  circle  c  be  denoted  by  M.     Then  at  every  point  of  c, 

mod  nz)  -  ■ r  -  — , 

and  consequently 

mod^  fi^z)dz  =  ^—J  ds  =  27tM, 

/dz 
z X-,  taken  around    any 

closed  curve  C  containing  the  point  a,  is  equal  to  zero,  except 
when  n  ^^  \,     When  /^  =  i,  this  integral  is  equal  to  2ni. 

For  the  value  of  the  integral  will  be  the  same  if  any 
circle  described  about  «^  as  a  center  be  taken  as  the  path  of 
integration.  Let  then  z  —  a  =  re'^,  where  r  is  a  constant  and 
0  varies  from  o  to  27C.     The  integral  becomes 


2jr     -  (n-l)i9 

e  de 


268  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

which  reduces  to  zero  except  when  n  =^  i,     li  n  =  i,  its  value 
is  2;r/,  whence 


P    dz 

^   z  —  a 


2m. 


Theorem  III. — If  fiz)  is  a  function  holomorphic  in  a  given 
region  S,  C  2,  closed  curve  the  interior  of  which  is  wholly 
within  5,  and  a  a  point  situated  within  (7,  then 


f  -^-dz  =  2nif{d). 


z  —  a 


For  describing  about  ^  as  a  center  a  small  circle  c  of  radius 
r,  the  equation 

*J^  z  —  a  ^'  z — a 

is  obtained.     But  at  every  point  of  c, 

f{z)=^f{d)  +  'n. 

where,  by  choosing  r  sufficiently  small,  the  modulus  of  t}  may 
be  made  less  than  any  fixed  positive  quantity.     Hence 

^cz—a  *J  c  z  —  a  ^cz  —  a 

but  by  the  preceding  theorems  the  first  term  of  the  right-hand 
member  is  equal  to  27tif{d),  and  the  second  term  is  equal  to 
zero. 

If  the  equation  of  the  theorem  just  established  be  differ- 
entiated with  respect  to  a,  the  following  important  formulas, 
expressing  the  successive  derivatives  of  a  holomorphic  function 
at  a  given  point,  are  obtained : 


1  .  2  .  .  .  «  frr^^^d,  =.  2nif<'\a). 

^ c\^z  ~  a) 


Art.  20.]  TAYLOR'S  series.  369^ 

The  integrals  in  the  first  members  of  these  equations  are  all 
finite  and  determinate  for  every  position  of  a  within  the  curve 
C.  Therefore  any  function  holomcrphic  in  a  given  region  ad- 
mits an  infinite  number  of  successive  derivatives  at  every 
interior  point.  Each  of  these  derivatives  being  monogenic 
must  be  continuous.     Hence  the  following: 

Theorem  IV. — If /(^)  is  holomorphic  within  a  given  region, 
there  exists  an  infinite  number  of  successive  derivatives  of 
f{z)^  which  are  all  holomorphic  within  the  same  region. 

Denote  by  r  the  shortest  distance  from  the  point  a  to  the 

curve  C,  Then  at  every  point  of  this  curve  \z  —  a\^r.  Let 
M  be  the  upper  extreme  of  the  modulus /(.s^)  on  (7,  and  /  the. 
length  of  C.     Then 


/»       fu\  ^   p   M    , 


^    ,.  =  Ml 


and  consequently  mod  /(**^  (a)  ^ . 


I  . 2  .  .  . «       Ml 

271 


In  particular,  if  6'  is  a  circle  having  a  for  its  center, 


mod/^»(.)-'-^--:"-^. 


Art.  20.    Taylor's  Series. 

Theorem. — Let/(.sr)  be  holomorphic  in  a  region  5,  and  let 
C  be  any  circle  situated  in  the  interior  of  5, 
If  a  be  the  center  and  a  Ar  t  any  other  point       /^  "^Np^ 

interior  to  C,  /  a+< 

Aa  +  0  -/(«)  +  tf\a)  +  y^/"(«)  + .  .  . 


270  FUNCTIONS   OF   A   COMPLEX    VARIABLE.  [CHAP.  VI. 

From  the  preceding  article,  denoting  a  variable  point  on  C 
byC, 

2711  J    ^  —  a  —  t 

I  .  i?  1.  ,  Z  .  .  t  ft 

where 

By  taking  n  sufficiently  great  the  modulus  of  R  may  be 
made  less  than  any  given  positive  quantity.  Let  Af  be  the 
upper  extreme  of  the  modulus  of  /(^)  on  the  circle  C,  p  the 
modulus  of  /,  and  r  the  modulus  of  C  —  <3:  or  radius  of  C.    Then 

'        ^27rJ^      r''^\r-p)     <  r  —  p\rl      * 

which,  since  p  <  r,  has  zero  for  its  limit  when  n  =  co . 
Writing  now  2  for  a-\-f,  Taylor's  Series  becomes 

The  series  is  convergent  and  the  equality  is  maintained  for 
-every  point  ^  included  within  a  circle  described  about  a  as  a. 
center  with  a  radius  less  than  the  distance  from  a  to  the  nearest 
critical  point  of  y"(^). 

When  a  is  equal  to  zero,  Taylor's  Series  takes  the  form 

/is)  =  /(o)  +  ^/'(o)  +  ^/"{o)  +...  +  _^/<-'(o)  +  .  .  .  , 
1.2  1.2  . . .  n 

expressing /(-sr)  in  terms  of  powers  of  ^.     This  form  is  known 

as  Maclaurin's  Series. 


Art.  21.]  Laurent's  series.  271 

Art.  21.    Laurent's  Series. 

Theorem. — Let  S,  a  porticn  of  the  ^-plane  bounded  by  two 
concentric  circles  C^  and  C^,  be  situated  in  the  interior  of  the 
region  E,  in  which  a  given  function  /{s)  is  holomorphic.  If  a 
be  the  common  center  of  the  two  circles,  and  a  -{-  ^  a.  point 
interior  to  Sy/{a-\-  t)  can  be  expressed  in  a 
convergent  double  series  of  the  form  - 

tn  =  00 

m  =  —  00 

With  a  -\-t  di's,  d.  center  construct  a  circle 
c  sufficiently  small  to  be  contained  within 
the  region  5.  If  then  C^  be  the  greater  of 
the  two  given  circles,  it  follows  from  Article  i8  that 

27ii^Cx  ^  —  a  —  t       27r/^Ca  C  —  ^  —  /       2;r2  '^c  C  —  «  —  ^* 
But  from  Article  19, 

2ni^<=  C-  a-  t      J^    ^  ^' 
whence 

•^^    ^   ^       2ni^^^  Z  -  a  -  t        27ti^c^  C -^  a  -  t 
The  two  integrals  of  the  right-hand  member  may  be  written 

where 


'  ~  27ti^^^r+'  (c  -  ^  -  /)  * 

But  |/|  <\Z  —  a\    at  every  point  of  C,,  and  |/|>  |C  —  a\  at 
every  point  of  (7,,  so  that  i^,  and  R^  both  have  zero  for  a  limit 


372  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

when  ;2  =  00  .  The  value  oifia  -\- 1)  can  therefore  be  expressed 
in  the  form 

f{a  +  t)  =  A,-{-  A,t  +  AJ'  -\-  A/  +  ... 

Since  in  the  region  5  the  function  f{z)/{z  —  <^)'^+'  is  holomor- 
phic  for  both  positive  and  negative  values  of  my  A^  may  be 
written 

^  _  JL  r    /(O    .. 

where  C  is  any  circle  concentric  with  C^  and  C^  and  included 
between  them. 

The  series  thus  obtained  is  convergent  at  every  point  a-\-t 
contained  within  the  region  S.  It  is  important  to  notice,  how- 
ever, that  when  the  positive  and  negative  powers  of  t  are  con- 
sidered separately,  the  two  resulting  series  have  different 
regions  of  convergence.  The  series  containing  the  positive 
powers  of  /  converges  over  the  whole  interior  of  the  circle  C^  \. 
while  the  series  of  negative  powers  of  t  converges  at  every 
point  exterior  to  the  circle  C^.  The  region  5  can  be  regarded, 
therefore,  as  resulting  from  an  overlapping  of  two  other 
regions  in  which  different  parts  of  Laurent's  Series  converge. 

Writing  z  iox  a  -{-  t,  Laurent's  Series  takes  the  form 

f(z)  =  ^.  +  Aiz  -  ^)  +  Ais  -df^... 

Consider  as  a  special  numerical  example  the  fraction 
"l = ^ ^       I         ^ 

(^  —    I)  (^  —  2)  (^-  3)        2(^-1)         ^  —  2^2(^  —   3) 

If  1^1  <  I,  all  three  terms  of  the  second  member,  when 
developed  in  powers  of  z,  give  only  positive  powers.  If 
I  <  1^1  <  2,  the  first  term  of  the  second  member  gives  a  series 
of  negative  descending  powers,  but  the  others  give  the  same  . 
series  as  before.  If  2  <  |^|  <  3,  the  first  and  second  terms- 
both  give  negative  powers.     If  \z\  >  3,  all   three  terms  give 


Art.  23.]  Fourier's  series.  273 

negative  powers,  and  the  development  of  the  given  fraction 
can  contain  no  positive  powers.  Thus  a  system  of  concentric 
annular  regions  is  obtained  in  each  of  which  the  given  frac- 
tion is  expressed  by  a  convergent  power-series.  Laurent's 
Series  gives  analogous  results  for  every  function  which  is  holo- 
morphic  except  at  isolated  points  of  the  ^-plane. 

Art.  22.    Fourier's  Series. 

Let  w  =  /{z)  be  holomorphic  in  a  region  S^ ,  and  let  it  be 
periodic,  having  a  period  equal  to  oo,  so  that  /{z  -\-  nao)  =  /(^), 
where  n  is  any  positive  or  negative  integer.  Denote  by  Sn  the 
region  obtained  from  S^  by  the  addition  of  noj  to  z ;  and  sup- 
pose that  the  regions  .  .  . ,  S_„ ,  .  .  . ,  5. , ,  5^ ,  5^ ,  .  .  . ,  5„  ,  ... 
meet  or  overlap  in  such  a  manner  as  to  form  a  continuous  strip 
S,  in  which,  of  course,  the  function  w  will  be  holomorphic* 
Draw  two  parallel  straight  lines,  incHned  to  the  axis  of  reals  at 
an  angle  equal  to  the  argument  of  cy,  and  contained  within  the 
strip  S.  The  band  T  included  between  these  parallels  will  be 
wholly  interior  to  S, 

2itiz 

By  means  of  the  transformation  z'  —  e  '^  the  band  T  in 
the  ^-plane  becomes  in  the  ^'-plane  a  ring  T'  bounded  by  two 
concentric  circles  described  about  the  origin  as  a  center,  z  and 
z  -\-  noa  falling  at  the  same  point  z'.  Since  w  is  holomorphic 
in  a  region  including  7",  and 

dw      dw  dz         oj     _  nrtz  dw 
dz'       dz  dz'       27ti        "    dz"* 

w  regarded  as  a  function  of  z'  will  be  holomorphic  in  T\ 
Hence,  by  Laurent's  Theorem, 

the  quantity  a  in  the  general  formula  of  the  preceding  article 
being  in  this  case  equal  to  zero.  Substituting  for  z'  its  value, 
the  preceding  equation  becomes 


274  FUNCTIONS   OF    A    COMPLEX    VARIABLE.  [ChaP.  VI. 

where 

In  the  latter  integral  the  path  is  rectilinear.  Denoting  its 
independent  variable  by  C  for  the  purpose  of  avoiding  confu- 
sion, the  value  of  w  becomes 


.   »«=co        ^^+w     2mTti  ^^     . 


^  »t=l    ^ 

m  =  l  => 

+  -  ^  sin  /      sin ^/(CKC 

w  =  l  ^ 


Art.  23.    Uniform  Convergence. 

Let  the  series  W=  «^o  +  ^i  +  ^a  +•  •  •  +  ^n  +  .  .  . ,  each 
term  of  which  is  a  function  of  z,  be  convergent  at  every  point 
of  a  given  region  S.  Denote  by  Wn  the  sum  of  the  first  n 
terms  of  W,  If  it  is  possible,  whatever  the  value  of  the  posi- 
tive quantity  e,  to  determine  an  integer  v,  such  that  whenever 
n  >  V 

\W-Wr.\<e 

at  every  point  of  S,  the  series  W  is  said  to  be  uniformly  con- 
vergent in  the  region  5. 

Uniformly  convergent  series  can  in  many  respects  be  treated 
in  exactly  the  same  manner  as  sums  containing  a  finite  number 
of  terms. 

Theorem  I. — A  uniformly  convergent  series,  the  terms  of 
which  are  continuous  functions  of  z,  is  itself  a  continuous 
function  of  z. 

For  at  any  point  z,  W  may  be  written  in  the  form 


Art.  23.]  UNIFORM    CONVERGENCE.  275 

IV  =  Wn  +  R;  and  at  a  neighboring  point  y,  W  =  Wn  +  R\ 
Hence 

W-    W   =    Wn-    Wn'  +R-R\ 

and  I  W-  W'\  =\  Wn  -  W^\-{-\R\-^\R'\, 

But  by  choosing  «  sufficiently  great,  \R\  and   \R'\  may  both 

be  made  less  than  any   given    positive    quantity  — .     Having 

chosen  n  thus,  W^  becomes  the  sum  of  a  finite  number  of 
continuous  functions.  It  is  then  continuous,  and,  by  making 
\z'  ~  z\  less  than  a  suitable  quantity  d,  \W  —  WJ\  may  be 

made  less  than  -.     But,  under  these  suppositions, 

\W-  W'\<e. 
W\Sy  therefore,  continuous  at  the  point  z. 

Theorem  H. — If  all  the  terms  of  a  uniformly  convergent 
series 

W=  w„  +  ze/,  +  .  .  .  +  2£;„  +  .  .  . 
are  continuous,  the  integral  of  the  series,  for  any  path  L  situ- 
ated in  the  region  of  uniform  convergence,  is  the  sum  of  the 
integrals  of  its  terms : 

For,  writing  W  =  Wn  +  R,  it  is  possible  to  choose  n  so  that, 
however  small  e  may  be,  |7?|  <  e  at  every  point  of  L,  If  n  be 
so  chosen, 

f  Wdz  =£  Wndz  +  £Rdz, 

But,  by  Article  15,  denoting  by  /  the  length  of  the  path  Z, 

mod    /  Rdz  <  e/, 

which,  when  n  =  co ,  has  zero  for  its  limit.     Hence 
f  Wdz  =  Vim   fWndz, 


276  FUNCTIONS   OF    A    COMPLEX    VARIABLE.  [ChAP.  VI 

Theorem  III. — If  the  series  IV  =  w^  +  w^-[- .  . .  +  ^k+  •  •  ► 
is  convergent,  and  the  series 

dz^  dz^         ^  dz^ 
is  uniformly  convergent  in  a  region  5,  and  if  further  the  terms 
of  W  are  continuous  in  S,  W  will  be  the  derivative  of  W, 


For,  integrating  W  from  a\.o  z  along  a  path  L  contained 
in  5, 

C  W'dz  =  wlz)  —  wj^d)  4-  .  .  .  +  Wn{z)  —  Wn{a)  +  .  .  . 
z=^W{z) -W{a), 

But  the  derivative  of  the  first  member  is  W\  which  must 
also  be  the  derivative  of  the  second  member,  and  therefore 
of  W. 

An  immediate  consequence  of  the  preceding  theorems  is 
the  following  : 

Theorem  IV. — If  the  terms  of  the  convergent  series 

IV  =  w,  -\-  w^  -\-  ,..-]-  'w„  -\- ..  . 

are  holomorphic  in  a  given  region  vS,  contained  in  the  region 
of  convergence,  and  if  the  series 

dz  ~^  dz  '^'"'^  dz  '^'  " 

is  uniformly  convergent,  IV  will  be  holomorphic  in  the  region 
5,  and  will  have  W  for  its  derivative. 

To  illustrate  by  an  example  that  uniformity  of  convergence 
is  essential  to  the  preceding  theorems,  take  the  series 

i-{-z  ^i{i+z»){i+z^+'y 

At  the  point  ^  =  i  each  term  is  continuous,  and  the  series 
is  convergent,  having  the  value  1/2.  The  series  is,  however,, 
discontinuous  at  ^  =  i.     For,  writing  it  in  the  form 

i-{-z       \i-\-z^       I  -{- z^    '    \i -{- z'       i-f^v    ' 


Art.  23.]  UNIFORM   CONVERGENCE.  277 

the  sum  of  the  first  n  terms  is 


I  -\-Z^ 


But  W  \s  the  limit  of  ^F„  when  «  =  oo ,  and  is  therefore 
unity  at  every  point  z  for  which  \z\  <  i,  and  zero  at  every 
point  for  which  \z\>  i. 

If  now  this  series  be  considered  for  the  points  within  and 
upon  a  circle  described  about  the  origin  as  a  center  with  an 
assigned  radius  less  than    unity,  the  remainder  after  ii  terms, 

or  I  —  Wn  = can,  by  a  suitable   choice   of  n,  be   made 

I  +  ^"  -^ 

iess  in   absolute   value  than   any  given   quantity.     In   such  a 

region,  then,  the  series  converges  uniformly,  and,  by  Theorem 

I,  can  have  no  point  of  discontinuity.     A  similar  result  holds 

for  the  region  exterior  to  any  circle  described  about  the  origin 

as  a  center  with  an  assigned  radius  greater  than  unity. 

By  means  of  Theorem  II  given  above  it  can  be  shown  that 
Laurent's  Series  is  unique.  For,  assuming  the  notation  used 
in  the  determination  of  the  series,  the  series  is  uniformly  con- 
vergent in  the  region  included  between  any  two  given  circles 
concentric  with  C^  and  C^ ,  both  being  interior  to  C^^  and  ex- 
terior to  C^.  '        > 

Suppose,  now,  that  two  such  series  are  possible  : 


m  =  CO 


/(a  +  /)  =  2    A^r  ■=  2  A  Jr. 

m  =  -  00  fn  =  —CO 

Divide  by  /"^S  and  integrate  along  any  circle  described  about 
a  as  a  center  and  included  in  the  region   of   uniform  converg. 

ence.     The  integral  j  t"^  -  »-  ^dt  for  such  a  path  is  zero,  except 

when  m  —  n\  the  integral  j  t'^dt  =  2i7t, 
Hence  for  such  a  path, 

ffSE±^=2inA„=2inA:; 


278  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [ChAP.  VL 

from  which  it  follows  that  A^  =  An,  and  the  two  series  are 
identical. 

Art.  24.    One-valued  Functions  with  Critical  Points. 

Theorem  I. — A  function  holomorphic  in  a  region  vS  and 
not  equal  to  a  constant,  can  take  the  same  value  only  at  iso- 
lated points  of  S. 

For  in  the  neighborhood  of  any  point  a  interior  to  S,  by 
Taylor's  theorem, 

f{B)  -f{a)  =  {z-  aV'ia)  +  ^I^  /"(a)  +  ... 

Unless /(^)  is  constant  over  the  entire  circle  of  convergence  of 
this  series,  the  derivatives  /'(a),  f'\d),  -  .  •  cannot  all  be 
equal  to  zero.  Let  f^^^Xa)  be  the  first  which  is  not  equal  to 
zero.     Then 

f(z)  -  /(a)  =  (z-ay[-^^!-^ \ f^'^y)        f^-a)+  .  .  .1 

-^ ^  ^     "^ ^  ^      ^         ^  \_i.2  .  ,  .  n  ^   1.2  .  .  .  {n-{-  ly-        ^  '         J 

If  \z — a\  be  given  a  finite  value  sufficiently  small,  the 
modulus  of  the  first  term  of  the  series  within  the  brackets  will 
exceed  the  sum  of  the  moduli  of  all  the  other  terms,  and  the 
same  result  will  hold  for  every  still  smaller  value  of  \3 — a\. 
For  values  of  s,  then,  distant  from  a  by  less  than  a  certain 
finite  amount, /(^)  —/{a)  is  different  from  zero. 

If,  on  the  other  hand,  the  function  is  constant  over  the  en- 
tire circle,  described  about  <3;  as  a  center,  within  which  Taylor's 
series  converges,  it  will  be  possible,  by  giving  in  succession 
new  positions  to  the  point  a,  to  show  that  the  value  of  the 
function  is  constant  over  the  whole  region  5. 

Theorem  II. — Two  functions  which  are  both  holomorphic 
in  a  given  region  vS  and  are  equal  to  each  other  for  a  system  of 
points  which  are  not  isolated  from  one  another,  are  equal  to 
each  other  at  every  point  of  5. 

For  let/(.s')  and  <p{ji)  be  two  such  functions.  By  the  pre- 
ceding theorem,  the  difference /(^)  —  0(^)  must  be  equal  to 
zero  at  every  point  of  5. 


Art.  24.]  ONE-VALUED    FUNCTIONS.  279 

Theorem  III. — A  function  which  is  holomorphic  in  every 
part  of  the  ^-plane,  even  at  infinity,  is  constant. 

For,  a  being  any  given  point,  whatever  the  value  of  ^, 

/(^)  =  At)  +  (^  -  tY\ci)  + . . .  +  r^=:^/""(«)  +  •  •  • 

But  by  Article  20,  r  being  the  radius  of  any  arbitrary  circle 
having   its  center  at  a,  and  J/ being  the  upper  extreme  of  the 
modulus  oi  f{z)  on  the  circumference  of  this  circle, 
,    ./„),  V  =1.2..  .  tiM 

But  M  is  always  finite,  and  r  may  be  made  indefinitely  great. 
Hence /"^"^(i^:)  =  o  for  all  values  of  n,  and 

Az)=:f{a). 

Theorem  IV. — If  a  function /(^),  holomorphic  in  a  region  5, 
is  equal  to  zero  at  the  point  a  situated  within  S,  the  function 
can  be  expressed  in  the  form 

f{z)  =  {z-  ar(P{z), 
where  m  is  a,  positive  integer,  and  (p[z)  is  holomorphic  in  5 and 
different  from  zero  at  a. 

For  in  the  neighborhood  of  the  point  a,  by  Taylor's  Theorem, 

/(«)=/('^)+(^ -«)/»  +  ••• 

Let/^*"^  (a)  be  the  first  of  the  successive  derivatives  at  ^  which 
is  not  equal  to  zero.     Then 

^^  ^   \_i  .2  .  .  .m   ^    1 .2  .  .  .{m-\-iy        '   ^ 

which  is  the  required  form.     The  point  «  is  a  zero  of /(^),  and 

m  is  its  order. 

Theorem  V. — If  the  point  ^  is  a  critical  point  of  a  given 
function /(^),  but  is  interior  to  a  region  5,  in  which  the  recip- 
rocal oi  f{z)  is  holomorphic,  the  function  can  be  expressed  in 
the  form 

where  m  is  a  positive  integer,  and  x{^)  is  holomorphic  in  the 
neighborhood  of  a. 


280  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

For  by  the  preceding  theorem 

^=(.-<0(.). 

where   (p{z)   is  holomorphic  and  not  equal  to  zero  at  z  ^=  a. 
Hence 

f(z^  =        '  -^  =       ^(^) 

•^  ^  ^       (z-a)^'  (p(z)         {z  -  a)"^' 

Further,  since  in  a  region  of  finite  extent  including  the 
point  a 

f(z)  =z    ^ h  .   .  .   +  — -  +  ^(^)y 

^^^        {z-aY'  ^  ^ z-a^^^'' 

a  being  an  ordinary  point  for  ^{z). 

The  point  <2  is  a  pole  o{  f{z)  and  7n  is  its  order. 

Theorem  VI. — A  function,  not  constant  in  value,  and  hav- 
ing no  finite  critical  points  except  poles,  must  take  values 
arbitrarily  near  to  every  assignable  value. 

For  suppose  that  f{z)  is  such  a  function,  but  that  it  takes 
no  value  for  which  the  modulus  oi  f{z)  —  ^  is  less  than  a  given 
positive  quantity  e.     Then  the  function 

I 
/«  -  A 
will  be  holomorphic  in  every  part  of  the  <s:-plane,  which,  by 
Theorem  III,  is  impossible  unless /{-s")  is  a  constant. 

Theorem  VII. — A  function  f{z),  having  no  critical  point 
except  a  pole  at  infinity,  is  a  rational  entire  function  of  z. 

For  the  only  critical  point  of  /(-)  is  a  pole  at  the  origin. 


Hence 


/0  =  ^  +  ---  +  T+^^)- 


where  (p(z)  is  holomorphic  over  the  entire  plane,  including  the 
point  at  infinity.  <p{z)  is  consequently  equal  to  a  constant  A^, 
The  given  function  therefore  can  be  written  in  the  form 


Art.  24.]  one-valued  functions.  281 

Theorem  VIII. — A  function /(<s)  whose  only  critical  points 
are  poles  is  a  rational  function  of  z,  ^  ~- 

The  poles  must  be  at  determinate  distances  from  one  an- 
other ;  otherwise  the  reciprocal  of  f{z)  would  be  equal  to  zero 
for  points  not  isolated  from  one  another.  The  number  of  poles 
<:annot  increase  indefinitely  as  |  ^  j  is  increased  ;  for  then  the 

reciprocal  of  /( -)  would  be  an  infinite  number  of  zeros  indefi- 
nitely near  to  the  origin.  The  total  number  of  poles  is  there- 
fore finite.  Let  a,  b,  ,  .  ,  denote  them.  In  the  neighborhood 
of  a  the  function  can  be  expressed  in  the  form 


{z  —  aY    '      '  '   ^  z  —a 
u  being  an  ordinary  point  for  0(^).     In  the  neighborhood  of  by 
0(<s)  can  be  expressed  in  the  form 

(73l)--  +  ---+ A  +  ^(^)' 

a  and  b  being  both  ordinary  points  for  ^{z).  Proceeding  in 
this  way  the  given  function  will  be  expressed  as  the  sum  of  a 
finite  number  of  rational  fractions  and  a  term  which  can  have 
no  critical  point  except  a  pole  at  infinity.  This  term  is  a 
rational  entire  function. 

Theorem  IX. — If  the  function  f{z)  has  no  zeros  and  no 
critical  points  for  finite  values  of  ^,  it  can  be  expressed  in  the 
form  f{2)  =  e^'\  where  ^(^)  is  holomorphic  in  every  finite  re- 
gion of  the  -S'-plane. 

f(z) 
For  -^  can  have  no  critical  points  except  at  infinity,  since 
y  w 

in  every  finite  region  of  the  ^-plane  f{z)  and  f\z)  are  holomor- 
phic and/(-8^)  is  different  from  zero.  Hence,  choosing  an  arbi- 
trary lower  limit  z^y  the  integral 


f 


is  holomorphic  in  every  finite  region.     The  function  f[z)  con- 
sequently must  take  the  form 


282  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VL 

where  g{z)  =  h{2)  +  log  f{z^. 

Theorem  X. — If  two  functions /(^)  and  ({){£)  have  no  criti- 
cal points  in  the  finite  portion  of  the^-plane  except  poles,  and 
if  these  poles  are  identical  in  position  and  in  order  for  the  two 
functions,  and  their  zeros  are  also  identical  in  position  and 
order,  there  must  exist  a  relation  of  the  form 

f{z)  =  cp{z)e^^^\ 

where  g{z)  is  holomorphic  in  every  finite  region  of  the  -s-'plane* 

For  the  ratio  of  the  two  functions  has  no  zeros  and  no 
critical  points  in  the  finite  portion  of  the  ^-plane. 

Art.  25.    Residues. 

If  a  one-valued  function  has  an  isolated  critical  point  ^,  it 
is  expressible  by  Laurent's  series  in  the  region  comprised  be- 
tween any  two  concentric  circles  described  about  a  with  radii 
less  than  the  distance  from  a  to  the  nearest  critical  point. 
Hence  in  the  neighborhood  of  a 

The   coefificient    of  (z  —  a)-^  in  this   expansion  is  called  the 
"residue"  of/(^)  at  the  point  a. 

If  any  closed  curve  6^  including  the  point  ^  be  drawn  in  the 
region  of  convergence  of  this  series,  and  f(z)  be  integrated 
along  C  in  a  positive  direction,  the  result  will  be 


/( 


{fz)dz  =  27rtB^. 
c 

The  following  may  be  regarded  as  an  extension  of  Cauchy*s 
theorem  : 

Theorem  I. — If  in  a  region  5  the  only  critical  points  of  the 
one-valued  function /(<a)  are  the  interior  points  a,  a\  .  .  .  ,  the 


Art.  25.J  RESIDUES.  28S 

integral   /  f(z)dz  taken  around  its  boundary  C  in  a  positive 
direction  is  equal  to 


S,A^¥^ 


27ti{B-\-B'-\-'"), 


where  B,  B\  .  .  .  are  the  residues  of  /(s)  at  the  critical  points. 
For  the  integral  taken  along  C  is  equal  to  the  sum  of  the 
integrals  whose  paths  are  mutually  exterior  small  circles  de- 
scribed about  the  points  ^,  a\  .  ,  . 

The  following  theorems  are  immediate  consequences  of  the 
preceding: 

Theorem  II. — If  in  a  region  having  a  given  boundary  (7  the 
only  critical  points  of  the  one-valued  function  /(^)  are  poles 
interior  to  C,  an  equation 

(Q^dz  =  2i7tlM-N) 
JcA^) 

exists,  M  denoting  the  number  of  zeros  and  N  the  number  of 
poles  within  C,  each  such  point  being  taken  a  number  of  times 
equal  to  its  order. 

For  in  the  neighborhood  of  the  point  a 

f{z)  =  {z-  ar<p{z) 

where  (p(z)  is  finite  and  different  from  zero  at  a,  and  m  is  a 

positive  integer  if  ^  is  a  zero,  a  negative  integer  if  ^  is  a  pole. 

Hence 

/\z)  ^     m  ^\z) 

Az)       z-a'^  cp{zy 

The  integrand,  therefore,  has  a  pole  at  every  zero  and  pole  of 
f{z),  and  its  residue  is  the  order,  taken  positively  for  a  zero, 
and  negatively  for  a  pole. 

Theorem  III. — Every  algebraic  equation  of  degree  n  has  n 
roots. 

For  let  f{z)  represent  the  first  member  of  the  equation 
5"  +  ^i-s^""'  +  .  .  .  +  ^«  =  O.     Since  f{z)   has   no   poles   in   the 


284  FUNCTIONS    OF    A    COMPLEX    VARIABLE,  [ChAP.  VI. 

finite  part  of  the  -s-'plane,  the  number  of  roots  contained  within 
any  closed  curve  C  will  be  given  by  the  integral 


(1\ 


But  taking  for  C  a  circle  described  about  the  origin  as  a 
center  with  a  very  great  radius,  this  integral  is 

I      rn^""-' -\- (n  —  i)a.2''-' + . . .   ,  i      f^nds,     ,     . 

where  e  has  zero  for  a  limit  when  1^|  =  oo .     Hence  the  limit 
of  the  preceding  integral,  as  |^|  is  increased,  is  ;/. 

Prob.  17.  Show  that  if  2:  =  00  is  an  ordinary  point  of /(-z),  that 
is,  \i  f{z)  is  expressible  for  very  great  value  of  2:  by  a  series  contain- 
ing only  negative  powers  of  z,  the  integral  oif{z)  around  an  infinitely 

great  circle  is  equal  to  27tt  into  the  coefficient  of  — .     This  coeffi- 
cient is  called  the  residue  for  s  =  00 . 

Prob.  18.  Show  that  the  sum  of  all  the  residues  of  /(z),  of  the 
preceding  problem,  including   the   residue  at  infinity,  is    equal    to 

^ero. 

(p(z)  .  .  .  . 

Prob.  19.  If  -j~r^  IS  a  rational  function  of  which  the  numerator 

is   of   degree  lower  by  2  than  the  denominator,  and  if  the  zeros 
a^y  a^f  ,  .  .  y  a„  of  the  denominator  are  of  the  first  order,  show  that 

^  (PM  _  ^ 


Art.  26.    Integral  of  a  One- valued  Function. 

It  was  shown  in  Article  18  that,  if  a  function /(^)  is  holo- 
tnorphic  in  a  given  region  5,  its  integral  taken  from  a  fixed 
lower  limit  contained  in  5  to  a  variable  upper  limit  S  is  a  one- 
valued  function  of  z  within  S.  If  F[z)  is  a  function  which 
takes  a  determinate  value  F{z^)  at  z  =  z„  and  is  one-valued 
while  z  remains  within  vS,  having  at  every  point  /{z)  for  its 
derivative,  the  integral  of  f{z)  from  z^  to  z  is  equal  to 
F(z)  —  F(z^).     If  F^{z)  is  another  function  fulfilling  these  con- 


Art.  26.]  INTEGRAL    OF    A    ONE-VALUED    FUNCTION.  285 

ditions,  so  that  the  integral  of  f{z)  can  be  written  also  in  the 
form  F^is)  —  F^{2^),  the  functions  F{z)  and  F^z)  differ  only  by 
a  constant  term  ;  for 

FAz)^F{z)^[,F,{z:i-F(z,)-]. 

Suppose  now  that/(-8:)  is  still  one-valued  in  S,  but  that  it 
has  isolated  critical  points  ^, ,  ^3,  .  .  .  interior  to  5".  Any  two 
paths  from  z^  to  Zy  which  inclose  between  them  a  region  con- 
taining none  of  the  points  ^,,  ^j,  .  . .,  will  give  integrals  identi- 
cal in  value.  Let  the  two  paths  Z, ,  L  include  between  them 
a  single  critical  point  a^]  and  consider  the  integrals  along 
these  two  paths.  The  integral  along  Z,  will  be  equal  to  the 
integral  along  the  composite  path  Z,Z~'Z,  where  the  exponent 
—  I  indicates  that  the  corresponding  path  is  reversed ;  for  the 
integral  along  L'^L  is  equal  to  zero.  But  L^L~^  is  a  closed 
curve,  or  "  loop,"  including  the  critical  point  a^^  and,  assuming 
that  it  is  described  in  a  positive  direction  about  a^,  the  inte- 
gral along  it  is  equal  to  27tiB^,  where  B^  is  the  residue  oi  f{z) 
at  a^..     Hence 

jf(z)dz  =  27tiB^-\-  jf{z)dz. 

If  now  the  two  paths  Z,,  Z  from  z^  to  z  include  between 
them  several  critical  points  a^,  ax,  a^,  .  .  .,  draw  intermediate 
paths  Z,,  .  .  .,  Z^,  so  that  the  region  between  any  two  consec- 
utive paths  contains  only  one  critical  point.  The  integral 
along  Z,  will  be  equal  to  the  integral  along  the  composite  path 
L^L~'L^  .  .  .  L^'^L^L'^L,  since  the  integrals  corresponding  to 
L~'L^,  .  .  .,  L^'^L^y  L~^L  are  all  equal  to  zero.  But  L^L~\ 
L^L^'\  .  .  .,  L^L'^  are  all  closed  paths  or  loops,  each  including 
a  single  critical  point,  so  that,  assuming  that  each  is  described 
in  a  positive  direction  and  that  B^,  B^,  ^^,  .  .  .  denote  the  resi- 
dues of  f{z)  at  the  critical  points, 

£^A^)dz  =  2rzi{B,  +  ^,  +  5^  +  .  .  .)  +  £/{e)dz. 

It  has  been  assumed  in  the  preceding  that  neither  of  the 
paths  Z,,  Z  intersects  itself.     In  the  case  where  a  path,  for 


286  FUNCTIONS   OF    A   COMPLEX   VARIABLE.  [Chap.  VI. 

example  L^,  intersects  itself  in  several  points  c^,  c^,  .  .  .,  it  is 
possible  to  consider  Z,  as  made  up  of  a  path  Z/  not  intersect- 
ing itself,  together  with  a  series  of  loops  attached  to  L/  at 
the  points  c^,  c^,  .  .  .  Each  of  these  loops  encloses  a  single 
-critical  point  a^  and,  if  described  in  a  positive  direction,  adds 
to  the  integral  a  term  2niB^  Each  such  loop  described  in  a 
negative  direction  adds  a  term  of  the  form  —  2niB^.  It  is  evi- 
dent that  the  form  of  each  loop  and  the  point  at  which  it  is 
attached  to  Z/  may  be  altered  arbitrarily  without  altering  the 
value  of  the  integral,  provided  no  critical  point  be  introduced 
into  or  removed  from  the  loop.  In  fact  all  the  loops  may  be 
regarded  as  attached  to  Z/  at  z^. 

It  can  be  proved  by  similar  reasoning  that  the  most  gen- 
eral path  that  can  be  drawn  from  z^  to  z  will  be  equivalent,  so 
far  as  the  value  of  the  integral  is  concerned,  to  any  given  path 
Z  preceded  by  a  series  of  loops,  each  of  which  includes  a  sin- 
gle critical  point  and  is  described  in  either  a  positive  or  nega- 
tive direction.  The  value  of  the  integral  is  therefore  of  the 
form 

/  f{z)dz  +  27ti{^m,B,  +  m,B,  +  ...), 
where  m^^  m^,  .  .  .  are  any  integers  positive  or  negative. 

r*  dz 

As  an  example  consider  the  integral  J^  .      The  only 

critical  point  \s  z  =z  a.  Any  path  whatsoever  from  z^  to  z  is 
equivalent  to  a  determinate  path,  for  example,  a  rectilinear 
path,  preceded  by  a  loop  containing  a  and  described  a  certain 
number  of  times  in  a  positive  or  negative  direction.  If  w  de- 
note the  integral  for  a  selected  path,  the  general  value  of  the 
integral  will  be  w  +  2n7ti.  If  now  a  straight  line  be  drawn 
joining  z^  to  a,  and  if  along  its  prolongation  from  a  to  infinity 
the  -sr-plane  be  cut  or  divided,  the  integral  in  the  ^-plane  thus 
divided  is  one-valued.  But,  with  the  variation  of  z  thus  re- 
stricted, any  branch  of  the  function  log  {z  —  a)  is  one-valued. 
Select  that  branch,  for  example,  which  reduces  to  zero  when 
z  =  a-\-  ^'     It  takes  a  determinate  value  for  z  —  z^,  and  its 


Art.  27.]  WEIERSTRASS'S   THEOREM.  287 

derivative  for  every  value  of  z  is .     Hence,  denoting  it 

by  Log  {z  —  a), 

J  — ^  =  Log  {z-d)-  Log  {z,  -a)  =  Log  ^^—^. 

For  a  path  not  restricted  in  any  way,  the  value  of  the  inte- 
gral is 

z  —  a 


/'    dz          _        z  —  a  .       , 
=  Log ±  2n7tt  =  log 
..z  —  a            ^  z,  —  a  -^  ^ 


z,  ~  a 


(piz) 
Prob.  2o.  If  -7-H  is  a  rational  function  of  z  of  which  the  numer- 
tp{z) 

ator  is  of  degree  lower  by  2  than  the  denominator,  and  if  the  zeros 
«i,  a^y  .  ,  .y  an  of  the  denominator  be  of  the  first  order,  show  that 
[*'    dz  :"  (piav)  ,       z  —  av 

/   —  ■^ iTr\  log » 

^z^z  —  a        1  t\)\av)         z^  —  aj 
where  2(p{ay)/il)\ay)  =  o.  (See  Prob.  18,  Art.  25.) 

Art.  27.    Weierstrass's  Theorem. 

Any  rational  entire  function  of  z,  having  its  zeros  at  the 
points  a^j  a^^  .  ,  .,  a,„,  can  be  put  in  the  form 

A{z-  a,p{z  -  a,Y^  .  .  .  {z  -  ^„)«-, 

where  A  is  2i  constant  and  n^,  n^,  .  .  .,  n,^  are  positive  integers. 
More  generally,  any  function  which  has  no  critical  point  in  the 
finite  portion  of  the  <s'-plane  and  has  the  points  a^,  .  .  .,  a^  as 
its  zeros,  is  of  the  form 

where  ^(<8r)  is  holomorphic  in  every  finite  region. 

The  extension  of  this  result  to  the  case  where  a  function 
without  finite  critical  points  has  an  infinite  number  of  zeros  is 
due  to  Weierstrass.  It  is  effected  by  means  of  the  following 
theorem  : 

Theorem. — Given  an  infinite  number  of  isolated  points  a^^ 


288  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [ChAP.  VI. 

a^y  .  .  .,  ^„,  .  .  .,  a  function  can  be  constructed  holomorphic  ex- 
cept at  infinity  and  equal  to  zero  at  each  of  the  given  points 
only.* 

For  the  given  points  can  be  taken  so  that 

\an\  increasing  indefinitely  with  ;2.  Consider  the  infinite  product 


0(^)  =  ^(i-^y»« 


where  /*„(^)  denotes  the  rational  entire  function 


Any  factor  may  be  written  in  the  form 


\         aj 


But  since 
lo 


[    ^^_   r  ^^  —  ^     _  "^^  -  r  ^""^^ 

^V  ~~  aj~  ~^^  an  —  z  ~  "~  ^  ""*  *  '~  na„''     "^^   a^^a^-zY 

the  path  of  integration  being  arbitrary  except  that  it  avoids 
the  points  a^,  a^,  .  .  .,  the  product  may  be  expressed  as 

r    z^dz 

ne'^n<^'\  in  which  tbJz)  =  —  /    —^, r. 

In  any  given  finite  region  of  the  ^--plane  it  will  be  possible 
to  assume  that  | -3" |  ^  p  <  <3;^  ,  since  \an\  increases  indefinitely 
with  n.     Divide  the  product  into  two  parts, 


nil  — Y^^'^\ne^^ 

I    \         aJ  m 


(*) 


The  second  part  is  equal  to 

m  • 

e 

*  The  following  proof  is  taken  from  Jordan,  Cours  d'Analyse,  2d  edition. 
Vol.  II. 


Art.  2/. J  WEIERSTRASS'S   THEOREM,  289 

Consider  the  series  2ipn{^)  and  ^tp„\2),  each  term  of  the  sec- 

m  m 

ond  being  the  derivative  of  the  corresponding  term  of  the  first. 
In  the  given  region 

1^/(^)1  = 


aj''(\a„,\-p)' 


a^\a^  —  z) 
Each  term  of  ^tp^'(s)  is  accordingly  less  in  absolute  value  than 

in 

the  corresponding  term  of  a  convergent  geometrical  progres- 
sion  independent  of  2.   The  series  ^ipjiz),  therefore,  converges 

m 

uniformly.     The  series  ^(p„{z)  also  converges,  since 


{•p 


K{z)\=  mod /' 4- J{sya  = 


where  /  denotes  the  length  of  the  path  of  integration. 

00 

By  Theorem  IV,  of  Article  23,  the  series  ^tpniz)  represents 

m 

in  the  given  region  a  holomorphic  function.     The  exponential 

m 

e 
also  must  be  holomorphic.     The  other  part  of  the  product 

containing  only  a  finite  number  of  factors  is  everywhere  holo- 
morphic, vanishing  at  all  of  the  points  a^,  a^,  .  .  .,  which  are 
situated  within  the  given  finite  region.  But  this  region  may 
be  extended  arbitrarily.  The  product  therefore  fulfils  the  re- 
quired conditions. 

In  the  preceding  demonstration  it  was  tacitly  assumed  that 
none  of  the  given  points  «,,  a.^,  .  .  .  was  situated  at  the  origin* 
To  introduce  a  zero  at  the  origin  it  is  necessary  merely  to  mul- 
tiply the  result  by  a  power  of  z. 

The  most  general  function   without  finite  critical   points 


290  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI, 

having  its  only  zeros  at  the  given  points  ^j,  «a,  .  .  .,  ^„  .  .  .,  can 
be  expressed  in  the  form 

where ^(^)  is  holomorphic  except  at  infinity;  for  the  ratio  of 
any  two  functions  satisfying  the  required  conditions  is  neither 
infinite  nor  zero  at  any  finite  point. 

By  means  of  Weierstrass's  theorem  it  is  possible  to  express 
any  function,  F{z\  whose  only  finite  critical  points  are  poles  as 
the  ratio  of  two  functions  holomorphic  except  at  infinity.  For, 
construct  a  function  ip{z)  having  the  poles  of  F{2)  as  its  zeros. 
The  product  F{2),  rp{z)  =  (p{z)  will  have  no  finite  critical  point. 
The  given  function  can,  therefore,  be  written 

which  is  the  required  form. 

In  applying  Weierstrass's  theorem  to  particular  examples, 
it  will  rarely  be  found  necessary  to  include  in  the  polynomials 
PJ2)  so  many  terms  as  were  employed  in  the  demonstration 
given  above.  It  is  quite  sufficient,  of  course,  to  choose  these 
polynomials  in  any  way  which  will  make  the  product  converge 
for  finite  values  of  ^  to  a  holomorphic  function.  Factors  of  the 
form  ,  . 

where  Pn{^)  is  chosen  in  such  a  manner,  are  called  "  primary 
factors." 

As  an  application  of  Weierstrass's  Theorem  take  the  reso- 
lution of  sin  2  into  primary  factors.  The  zeros  of  sin  s  are  o, 
±7r,  ±27r,  .  .  .,  ±n7ry   ....     Consider  factors  of  the  form 

z 

(  2  \  i^ 

\  HTt' 

SO  that  Pni^)  contains  only  one  term  — ,  and 

nTT 

,  ,  .  r^        zdz 


f     nninn 

0         ^ 


Art.  27.J  WEIERSTRASS'S   THEOREM.  291 

The  series  2tp„\z)  will  converge  uniformly  in  any  region  at 

tn 

every  point  of  which  \z\  =  p  <  mn  ;  for,  since 


nni^in  —  z) 


\  \mt\l  \         mni 


\nn\l  \         m: 

each   term   is  less  in  absolute  value  than  the  corresponding 
term  of  the  series 


\         mnl 


A  similar  result  holds  for  the   series  2tpn{z)'     The  two 

-m 

series 

tn  -m 

are  also  convergent;  for  |^„(^)|  cannot  exceed  the  upper  ex- 
treme of  \ipj{z)\  multiplied  by  /,  the  length  of  the  path  of 
integration  from  the  origin  to  the  point  2.  These  series 
accordingly  represent  holomorphic  functions  in  the  region  for 
which  1^1  =  p.     Hence  the  expression  required  is 

sin<s:  =  e^'^nii ^V*^- 

-00  \         nn) 

It  will  be  shown  in   the  course  of  the   next  article  that 

^')=  I. 

Prob.  21.  If  ft? J  and  ft?,  be  two  quantities  not  having  a  real  ratio, 

the  doubly  infinite  series  of  which  the  general  term  is  7 ; rr 

is  absolutely  convergent  if  /  >  2.     Hence  show  that  the  product 


\e 


where  go  =  mGO^-\-  nao^ ,  defines  a  holomorphic  function  in  any  finite 
region  of  the  -s-plane.  This  function  is  Weierstrass's  sigma  func- 
tion, and  is  the  basis  of  his  system  of  elliptic  functions. 


292  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [ChAP.  VL 

Art.  28.     Mittag-Leffler's  Theorem. 

Any  one-valued  function  f{z)  with  isolated  critical  points 
<3:, ,«,,..  .  can  be  represented  in  the  neighborhood  of  one  of 
these  points  by  Laurent's  series  ;  viz. : 

Az)  =  A,^  A^z  -  a:)  +  A^{z  -  a^y  +  .  .  . 

+  Biz  -  a^Y'  +  Biz  -  a„)-^  +  .  .  .  . 

Hence  /{zf  =  cp(z)  +  Gn{^^), 

where  (p{z)  is  holomorphic  in  a  region  containing  the  point  a„ , 
and  G{- 1  is  holomorphic  over  the  whole  plane  excluding 

the  point  a„.     If  a„  is  a  pole  oi/{z),  GA j  contains  a  finite 

number  of  terms  ;    otherwise  it  is   an  infinite   series.      If  the 

number  of  critical  points  is  finite,  and  the  function  Gn[ ) 

is  formed  at  each  such  point,  by  subtracting  the  sum  of  these 
functions  from  /{z)  a  remainder  will  be  obtained  which  has  no 
critical  point  in  the  finite  part  of  the  plane.  This  remainder 
can  be  expressed  as  a  series  of  ascending  powers  G{z)  converg- 
ing for  every  finite  value  of  z.  The  function  /(z)  can  there- 
fore be  written  in  the  form 

analogous  to  the  expression  of  a  rational  function  by  means  of 
partial  fractions. 

The  extension  of  this  result  to  the  case  where  the  number 
of  critical  points  is  infinite  is  due  to  Mittag-Leffler.  Let  a^, 
^,,  .  .  .,  ^«,  .  .  .  be  the  critical  points  of  the  one-valued  func- 
tion /(z)f  and  suppose  that 

\a„\  increasing  without  limit  when  n  is  increased  indefinitely. 

Let,  further,  gJ )  be  the  sen'es  of  negative  powers  of 

\2  —  a„f 


Art.  38.]  mittag-leffler's  theorem.  293 

z — a^  contained  in  the  expansion  oif{z)  according  to  Laurent's 
Series  in  the  neighborhood  of  a^. 

The  function  g1 j,  having  no  critical  point  except  at 

an,  may  be  developed  by  Maclaurin's  series  in  the  form 

^"(rr^)  =  ^0^"'  +  ^'^'^  +  •  •  •  +  ^"'"^  +  •  •  •  ' 

and  the  series  will  converge  uniformly  within  a  circle  described 
about  the  origin  as  a  center  with  any  determinate  radius 
Ph  <  |^«|-     Within  the  same  circle  Maclaurin's  series,  applied 

to  Gni K  the  derivative  with  respect  to  z  of  GJ 1, 

\z  —  a  J  ^  V^  —  a„) 

converges  uniformly.     Hence,  for  any  point  within  the  circle 

Fn{3)  representing  the  first  v  -\-  I  terms  of  the  development  of 

GJ 1  by  Maclaurin's  theorem,  F'(z)  its  derivative,  and 

\z  —  aj     ' 

R,  R'  remainders  which  by  a  suitable  choice  of  v  may  be  made 
less  in  absolute  value  than  any  given  quantity. 

Choose  the  positive  quantities  £^,  E^,  .  .  .,  E„,  .  .  .  so  that 
the  series  E, -\-  E^-\-  ,  .  .  -\-  E„-\-  .  .  ,  is  convergent.  Choose 
also  in  connection  with  each  of  the  points  ^,,  <2,,  .  .  . ,  ^„,  .  .  ., 
an  integer  v  such  that 

i^  kl  <P»  <  I^J;  and,  in  general, 


294  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [ChAP.  VI. 

Consider  now  the  series 

in  any  finite  region  of  the  plane,  the  points  a^,  a^, .  .  .  ,  a^,  .  ,  T 
being  excluded.  Since  \a^\  increases  indefinitely  with  n,  it  is 
possible,  in  any  finite  region  of  the  <s'-plane,  to  assume  that 
\^\  </3w<  \^m\'  Separate  from  each  of  these  two  series  its 
first  m  —  \  terms.  These  terms  will  have  in  each  case  a 
finite  sum.  The  remaining  terms  of  either  series  taken  in  order 
will  be  less  in  absolute  value  than  E^,  -£",„  + 1,  •  .  •  respectively, 
\z\  being  less  than  each  of  the  quantities  p^,  /c?„,+  i, .  .  .  .  Ac- 
cordingly, each  of  the  series 
I 


Fi.<A,  mG,:[^yFA.^ 


\z  —  a^ 

is  absolutely  convergent  for  every  value  of  z  except  ^, ,  ^j, .  . . , 
a^y  ,  .  .  .  It  is  evident,  further,  that  in  any  given  finite  region, 
from  which  the  points  a^,  a^,  .  .  . ,  a^,  >  .  -  are  excluded,  the 
two  series  converge  uniformly.  In  such  a  region  any  term  of 
either  series  is  holomorphic ;  and,  therefore,  by  Theorem  IV 
of  Article  23,  the  first  of  these  series  defines  a  holomorphic 
function. 

The  point  a^  is  an  ordinary  point  for  the  difference 

/^-)  -  \A-^-  ^"(^J = \i^'^  -  ^-t^J]  +  ^"(^)' 

since  in  its  neighborhood  this  difference  may  be  developed  as 
a  convergent  series  containing  only  positive  powers  of  ^  —  a^. 
In  the  same  way  each  of  the  points  <2,,  ^3,  .  .  . ,  <3;„,  .  .  .  is  an 
ordinary  point  for  the  function 

This  function,  therefore,  can  have  no  critical  point  except  at 
infinity,  and  must  be  expressible  as  a  series  G{z)  containing 
only  positive  powers  of  z  and  converging  uniformly  in  any 
finite  region  of  the  ^-plane.  Hence  the  function  f{z)  may  be 
put  in  the  form 


Art.  28.]  mittag-leffler's  theorem.  295 

I       \ 


f{z)=G{z)Jr 


.(^)], 


1 

in  which  the  character  of  each  critical  point  is  exhibited. 

As  an  application  of  Mittag-LefHer's  theorem  consider  cot  z. 
Its  critical  points  are  ^  =  o,  ±  ;r,  ±  2;r,  ....  In  the  neigh- 
borhood of  -a  —  o,  cot  <8^  —  —  is  holomorphic ;  and  in  the  neigh- 

2 

borhood  of  ^  =  nn,  n  being  any  positive  or  negative  integer, 

cot  2 is  holomorphic.     The  series 

2  —  nn 

+  00  T 


2—  nn 


in  which  m  is  an  arbitrary  positive  integer,  is  not  convergent 
for  finite  values  of  2,  even  when  \2\<im.     The  series 


^[.2  —  n7t~^  nnj  ~~  ^  n7ti2  —  nn)  ~~  ^         (  2\ 

\  nnl 

is,  however,  absolutely  convergent  at  every  point  for  which 
1^1  <  w.     For  the  modulus  of  any  term  is  equal  to 

1^1  _         \A 


\  nnl 


nn  I  \  nn> 


and,  therefore,  less  than  the  corresponding  term  in  the  series 


L^\        m       ft 

mn) 
A  similar  result  holds  for  the  series 


V.2  +  nn       nnj 


It  IS  easy  to  see  now  that  the  reasoning  employed  in  the 
demonstration  of  Mittag-Lefifler's  theorem  may  be  applied  to 
show  that  the  series 


296  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [ChAP.  VI. 

where  the  summation  does  not  include  n  =  o,  defines  a  func- 
tion holomorphic  in  any  finite  region  of  the  ^-plane,  the  points 
o,  ±  ;r,  ±  2;r, .  . .  being  excluded.     The  difference 

cot  ^ ^ 

can  have  no  critical  point  except  at  infinity.  It  must,  there- 
fore, be  expressible  as  a  series  G{2)  of  positive  powers  of  js, 
having  an  infinite  circle  of  convergence.     Hence 

cot  2 = Giz) + - + ^  r — 5 — h  - 1- 

-  00      I—  — ' 

The  next  step  is  to  determine  G{z),  It  is  to  be  observed 
that,  if  G{z)  is  a  constant,  its  value  must  be  zero,  since 
cot  (—  ^)  =  —  cot  z.  If  G{2)  is  not  a  constant,  differentiation 
of  the  preceding  expression  for  cot  z  gives 

I  I        jL°°  t 


It  follows,  by  changing  z  into  z  -\-  rty  that 

G'{z  +n)=  G'{z\ 

Hence  G'{z)  is  periodic,  having  a  period  equal  to  n ;  and  as  the 
point  z  traces  a  line  parallel  to  the  axis  of  reals,  G'{z)  passes 
again  and  again  through  the  same  range  of  values.  But  G'{z), 
being  the  derivative  of  G{z),  is  holomorphic  for  every  finite 
value  of  z.  It  can,  therefore,  become  infinite,  if  at  all,  only 
when  the  imaginary  part  of  z  is  infinite.  If  z  be  written  in 
the  form  x-{-  iy,  the  value  of  G'{z)  may  be  expressed  as 

, I         _i_^  \ /   2iey{cos  X  -\-  i  sin  x  V 

^  ^  ~  {x  -\-  iff      ^  {x^iy  —  nnf      \(cos  2x-\-i  sin  2x)  —  e^y/' 

When  jj/  =  ±  00    the    first   and    last   terms  of  the   second 
member  vanish.     In  regard  to  the  series  it  can  be  proved  that, 


Art.  28.]  mittag-leffler's  theorem.  297 

for  any  given  region  is  which  y  is  finite  and  different  from 
zero,  an  integer  v  can  be  found  such  that  the  sum  of  the  moduli 
of  tl>ose  terms  for  which  \n\>  r  is  less  in  absolute  value  than 
any  previously  assigned  quantity  e.  As  \y\  is  increased  the 
modulus  of  each  of  these  terms  is  diminished.  The  modulus 
of  their  sum,  therefore,  cannot  exceed  e  when  j  =±00.  But 
when_;'=  ±00  the  sum  of  any  finite  number  of  terms  of  the  series 
is  zero.  Hence  the  limit  of  the  whole  series  is  zero.  G'(s), 
therefore,  never  becomes  infinite.  Hence,  by  Theorem  HI, 
Article  24,  it  is  constant,  and  is  equal  to  zero.  It  follows  that 
G{z)  is  equal  to  zero. 

The  expression  for  cot  z  is  accordingly 


nit         HTTj 

The  logarithmic  derivative  of  the  product  expression  for 
sin  s,  given  in  the  preceding  article  as  an  example  of  Weier- 
strass's  theorem,  is 

cot^=^'(^)+- +^r — 5 — h--]- 

—  00 

Hence  g{z)  in  that  expression  is  a  constant.  Makings  =  O, 
its  value  is  seen  to  be  unity. 

Prob.  22.  From  the  expression  for  cot  z  deduce  the  equation 

+  00 

COSec''  2:   =    ^  , Ta, 

where  the  summation  does  not  exclude  ;/  =  o. 
Prob.  23.  Show  that  the  doubly  infinite  series 
I    .    ^- 


:^)  = 


L(^ 


-1 -L] 

-  coy    Gj\y 


where  00=  moo^  -\-  nao^ ,  defines  a  function  whose  only  finite  critical 
points  are  z  =  00.  This  function  is  Weierstrass's  ^-function.  (Com- 
pare Problem  21.) 

Prob.  24.  Prove  that 


FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [Chap.  VI. 

Prob.  215.  Prove  that  «)'(s)  =  —  2^7 r-^,  where  the  summa- 

tion  does  not  exclude  09  =  o.  > 


Art.  29.    Critical  Lines  and  Regions. 

The  functions  whose  properties  have  been  considered  in  the 
preceding  articles  have  been  assumed  to  have  only  isolated 
critical  points.  That  an  infinite  number  of  critical  points  may 
be  grouped  together  in  the  neighborhood  of  a  single  finite 
point  is  evident,  however,  from  the  consideration  of  such  ex- 
amples as 

w  =  cot  - ,         w  =  ^^°s^<=  jr^. 
z 

In  the  former  an  infinite  number  of  poles  are  grouped  in  the 
neighborhood  of  the  origin.  In  the  latter  an  infinite  num- 
ber of  essential  singularities  are  situated  in  the  vicinity  of  the 
point  z  ^=  a. 

It  is  easy  to  illustrate  by  an  example  the  occurrence  of  lines, 
and  regions  of  discontinuity.     Take  the  series^ 

The  sum  of  its  first  n  terms  is 


2       —  I 

vi^hich  converges  to  unity  if  |-s'|<  I,  and  to  zero  if  \z\>  I. 
Hence  the  circle  |^|=i  is  a  line  of  discontinuity  for  this 
series. 

Consider  now  any  two  regions  5,  and  5,,  the  former  situated 
within,  the  latter  without,  the  unit  circle.  Let  <p{s)  and  ^{2) 
be  two  arbitrary  functions  both  completely  defined  in  these 
regions.     The  expression 

*  This  series  is  due  to  J.  Tannery.  See  Weierstrass,  Abhandlungen  aus  der 
Funcdonenlehre  (1886),  p.  102. 


Art.  29.]  critical  lines  and  regions.  290 

will  be  equal  to  (p{z)  in  S^  and  ipiz)  in  S^.  In  regions  com- 
pletely separated  from  one  another  by  a  critical  line,  the  same 
literal  expression  may  thus  represent  entirely  independent 
functions. 

For  a  single  continuous  region,  however,  in  the  interior  of 
which  exist  only  isolated  critical  points,  the  character  of  the 
function  in  one  part  determines  its  character  in  every  other 
part.  Let  5  be  such  a  region,  and  assume  that  its  boundary  is  a 
critical  line.  In  the  neighborhood  of  any  interior  point  ^,  not 
a  critical  point,  the  given  function  is  expressible  as  a  power 
series,  viz. : 

%)  =  /(«)  +  (^  -  a)f'{a)  + .  .  .  +  JlZf )!/<.)(<,)  + .  .  . 

1  .  ^   ...    72 

This  series  will  converge  uniformly  over  a  circle  described 
about  ^  as  a  center  with  any  determinate  radius  less  than  the 
distance  from  a  to  the  nearest  critical  point.  It  serves  for  the 
calculation  oi  f{z)  and  all  its  successive  derivatives  at  any  point 
b  interior  to  this  circle.  From  the  preceding  power  series,  ac- 
cordingly, can  be  obtained  another 

M  =  M + (^  -  iV'{6)  +  ...  +  lizillyx-x^)  + . . . , 

representing  the  /(^)  within  a  circle  described  about  /^  as  a 
center.  In  general,  the  point  d  can  be  so  chosen  that  a  portion 
of  this  new  circle  will  lie  without  the  circle  of  convergence  of 
the  former  power  series.  At  any  new  point  c  within  the  circle 
whose  center  is  d,  the  value  of  the  function  and  all  its  succes- 
sive derivatives  can  be  calculated  ;  and  so,  as  before,  a  power 
series  can  be  obtained  convergent  in  a  circle  described  about  c^ 
as  a  center  and,  in  general,  including  points  wot  contained  in 
either  of  the  preceding  circles.  By  continuing  in  this  manner 
it  will  be  possible,  starting  from  a  given  point  a  with  the  ex- 
pression oi  f{z)  in  ascending  powers,  to  obtain  an  expression  of 
the  same  character  at  any  other  point  k  which  can  be  connected 
with  ^  by  a  continuous  line  everywhere  at  a  finite  distance 
from  the  nearest  critical  point.     It  follows  that  the  character  of 


300  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [ChaP.  VI. 

the  function  everywhere  within  S  can  be  determined  completely 
from  its  expression  in  ascending  power  series  in  the  neighbor- 
hood of  a  single  interior  point. 

It  will  be  impossible  by  the  process  just  explained  to  derive 
any  information  in  regard  to  the  function  at  points  exterior  to 
S.  The  example  given  above,  furthermore,  shows  that  a  com- 
plete definition  o( /{s)  within  5  may  carry  with  it  the  definition 
of  an  entirely  independent  function  without  5. 

As  an  example  of  a  function  having  a  critical  region  con- 
sider the  function  defined  by  the  series 

I -\- 22  ~\- 2Z*  -\- 22^  -[- .  .  .  , 

which  represents  a  function  without  critical  points  in  the 
interior  of  the  circle  |^|  =  i.  For  points  on  or  without  this 
circle  the  series  is  divergent ;  and,  further,  it  is  impossible  to 
obtain  from  it  an  expression  converging  when  l^s"!  =  i.  The 
function  thus  defined,  consequently,  exists  only  in  the  region 
interior  to  the  unit  circle.     By  changing^  into  i/z  a  series 

2         2  2 

is  obtained,  representing  a  function  which  has  no  existence  in 
the  interior  of  the  unity  circle.  P'unctions  in  connection  with 
which  such  regions  arise  are  called  "  lacunary  functions."* 


Art.  30.    Functions  Having  n  Values. 

Let  the  function  w  =/(2)  take  at  the  point  2^  of  a  given  re- 
gion vS  a  value  w^^K  Suppose  that  along  any  continuous  path, 
beginning  at  2^,  and  subject  only  to  the  conditions  that  it  shall 
remain  in  the  interior  of  5  and  shall  not  pass  through  certain 
isolated  points  ^, ,  ^, ,  .  .  .  ,  w  is  continuous  and  has  a  contin- 
uous derivative.  If  it  is  impossible,  when  2  traces  such  a  path, 
to  return  to  the  point  2^  so  as  to  obtain  there  a  value  of  w  dif- 
ferent from  'it/'^\  w  is  one-valued  in  the  region  5.     On  the  other 

*  Poincar6,   American  Journal  of   Mathematics,  Vol.  XIV;  Harkness  and 
Morley,  Theory  of  Functions  (1893),  p.  119 


Art.  30. J  FUNXTIONS    HAVING    //    VALUES.  301 

hand,  certain  paths  may  lead  back  to  z^  with  new  values  of  w. 

Suppose  that  at  each  point  of  5,  except  a^^  a^,  ,  .  .  ^  w  has 
n  different  values,  and  that  starting  from  such  a  point  5"^  and 
tracing  any  continuous  curve  not  passing  through  a^  ,  a^^  .  .  .  , 
the  several  values  of  zu  give  rise  to  n  branches  Wj ,  w, ,  .  .  .  ,  w^y 
each  of  which  is  characterized  by  a  continuous  derivative.  In 
the  neighborhood  of  a^  any  one  of  the  points  ^j ,  ^^ ,  .  .  . 
these  branches  are  said  to  be  distinct  or  not,  according  a?  small 
closed  curves  described  about  this  point  lead  from  each  value  of 
w  back  to  the  same  value  again,  or  cause  some  of  the  branches 
to  interchange  values.  In  the  latter  case  the  point  is  a  branch 
point. 

About  any  branch  point  Uk  as  a  center  describe  a  small  cir- 
cle ;  and  suppose  that,  starting  from  any  point  of  it  with  the 
value  Wa  corresponding  to  a  certain  branch,  the  values 
w^  ^Wy  ...  are  obtained  by  successive  revolutions  about  Uk , 
the  original  value  being  reproduced  after  p  revolutions.  In- 
troduce now  a  new  independent  variable  2'  such  that 

z'  =  {z-  a,)'K 
It  can  be  shown  that  when  z  makes  one  revolution  about 
Uj^y  z'  makes  only  one  /th  part  of  a  revolution  about  the  ori- 
gin of  the  y-plane,  and  that  to  a  complete  revolution  of  z 
about  the  origin  of  the  .s^'-plane  correspond  /  revolutions  of  z 
about  a^.  Considering  then  the  branch  w^,  as  a  function  of  z' , 
the  origin  cannot  be  a  branch  point,  for  whenever  z'  describes 
a  small  circle  about  it,  the  value  Wo.  is  reproduced.  The 
branch  Wa  must  accordingly  be  expressible  by  Laurent's 
series  in  the  form 

—  00 
or,  substituting  for  z'  its  value, 

1  2 

w^  =  A,-^AXz-a,)'-\-A(,z-a,y-\-.,. 

1  _2 

This  expression  makes  plain  the  relation  between  the  different 


302  FUNCTIONS    OF    A    COMPLEX    VARIABLE.  [ChaP.  VI. 

branches  of  a  function  in  the  neighborhood  of  a  branch  point. 
When  the  development  of  a  branch  in  the  neighborhood  of  one 
of  its  branch  points  gives  rise  to  only  a  finite  number  of  terms 
containing  negative  powers,  the  branch  point  is  called  a  ''  polar 
branch  point." 

Consider  the  functions 

P^  =^  w^  +  w,  +  .  .  .  +  w„  , 

P^    =    W^W^   +   W^W^    —    .   .    .   -\-    Wn-.Wny 


P„   =    W,W^   .   .    .    Wn. 

Each  of  these  functions  is  unchanged  in  value  when  several  or  all 
of  the  quantities  w„  w^,  .  .  ,  ,  w„  are  interchanged,  and  is  con- 
sequently  a  one-valued  function  of  ^  within  5.  Hence  w  must 
satisfy  an  equation  of  the  nth  degree, 

w**  +  P.w''-'  +  P,  w**-^  +  .  .  .  +  T'*'  =  o, 

the  coefificients  of  which  are  one-valued  functions  of  2  having 
only  isolated  critical  points  within  5.  When  the  entire  ir-plane 
can  be  taken  as  the  region  S,  and  those  branch  points  at  which 
the  branches  do  not  all  remain  finite  are  polar  branch  points, 
the  only  other  critical  points  being  poles  for  one  or  more 
branches,  the  functions  P,,  P^,  .  .  .  ,  P„  are  rational  functions 
of  2.     In  this  case  w  is  an  algebraic  function  of  2. 


Art.  1.]  EQUATIONS   OF    FIRST    ORDER    AND    DEGREE.  303 


Chapter  VII. 

DIFFERENTIAL    EQUATIONS. 
By  W.  Woolsp:y  Johnson, 

Professor  of  Mathematics  in  the  U.  S.  Naval  Academy. 

Art.  1.    Equations  of  First  Order  and  Degree. 

In  the  Integral  Calculus,  supposing/  to  denote  an  unknown 
function  of  the  independent  variable  x,  the  derivative  of  y  with 
respect  to  x  is  given  in  the  form  of  a  function  of  x,  and  it  is 
required  to  find  the  value  of  /  as  a  function  of  x.  In  other 
Avords,  given  an  equation  of  the  form 

%=A^),     or    dy  =  A^)dx,  (i) 

of  which  the  general  solution  is  written  in  the  form 

y  =  ff(x)dx,  (2) 

it  is  the  object  of  the  Integral  Calculus  to  reduce  the  expres- 
sion in  the  second  member  of  equation  (2)  to  the  form  of.  a 
known  function  of  x.  When  such  reduction  is  not  possible, 
the  equation  serves  to  define  a  new  function  of  x. 

In  the  extension  of  the  processes  of  integration  of  which 
the  following  pages  give  a  sketch  the  given  expression  for  the 
derivative  may  involve  not  only  x,  but  the  unknown  function 
y ;  or,  to  write  the  equation  in  a  form  analogous  to  equation 
(i),  it  may  be 

Mdx  +  Ndy  =  o,  (3) 

in  which  M  and  N  are  functions  of  x  and  y.  This  equation  is 
in  fact  the  general  form  of  the  differential  equation  of  the  first 
order  and  degree;  either  variable  being  taken  as  the  independ- 
ent variable,  it  gives  the  first  derivative  of  the  other  variable 


304  DIFFERENTIAL    EQUATIONS.  [ChaP.  VII.. 

in  terms  of  x  and  y.  So  also  the  solution  is  not  necessarily  an 
expression  of  either  variable  as  a  function  of  the  other,  but  is 
generally  a  relation  between  x  and  y  which  makes  either  an 
impHcit  function  of  the  other. 

When  we  recognize  the  left  member  of  equation  (3)  as  an 
"exact  differential,"  that  is,  the  differential  of  some  function  of 
X  and  J,  the  solution  is  obvious.  For  example,  given  the  equa- 
tion 

xdy  +  ydx  =  o,  (4) 

the  solution  xy  =  C,  *  (5) 

where  C  is  an  arbitrary  constant,  is  obtained  by  "  direct  inte- 
gration." When  a  particular  value  is  attributed  to  (7,  the  result 
is  a  "  particular  integral  ;  "  thus  jj/  =  x~^  is  a  particular  integral 
of  equation  (4),  while  the  more  general  relation  expressed  by^ 
equation  (5)  is  known  as  the  "  complete  integral." 

In  general,  the  given  expression  Mdx  -\-  Ndy  is  not  an  ex- 
act differential,  and  it  is  necessary  to  find  some  less  direct 
method  of  solution. 

The  most  obvious  method  of  solving  a  differential  equation 
of  the  first  order  and  degree  is,  when  practicable,  to  "  separate 
the  variables,"  so  that  the  coefificient  of  dx  shall  contain  x 
only,  and  that  of  dy,  y  only.     For  example,  given  the  equation 

( I  —  y)dx  +  (I  +  x)dy  =  o,  (6) 

the  variables   are   separated    by  dividing  by  {i -{- x){i  —  y).. 

Thus  : h  — =^—  =  o. 

I  -j-  jir  '   I  —  y 

Each  term  is  now  directly  integrable,  and  hence 
log  (I  +^)  -  log  (I  -y)  =  c. 

The  solution  here  presents  itself  in  a  transcendental  form, 
but  it  is  readily  reduced  to  an  algebraic  form.  For,  taking  the 
exponential  of  each  member,  we  find 

I  A-  X 

—L—  —  e'  —  C,    whence    i  -i-  ;ir  =  ^(l  —  y\  (;> 

where  C  is  put  for  the  constant  ^. 


Art.  2.]  GEOMETRICAL    REPRESENTATION.  305 

To  verify  the  result  in  this  form  we  notice  that  differentia- 
tion gives  dx  ^=  —  Cdy,  and  substituting  in  equation  (6)  we  find 

-  C{i  —y)-{-i+x  =  o, 

which  is  true  by  equation  (7). 

Prob.  I.  Solve  the  equation  dy  -\-y  tan  x  dx=-  o. 

(Ans.  y—C  cos  x^ 

Prob.  2.  Solve  ^  +  ^V  =  a\  f  Ans.  ^1:^  =  ^^2a*-.\ 

dx  ^  \         by  —  a  ) 

Prob.  3.  Solve  f  =  4^.  (Ans.  y  =  i^A) 

"^  dx        X  -\-  1  \  -^       1  —  ex  I 

Prob.  4.  Helmholtz's  equation  for  the  strength  of  an  electric 
current  C  at  the  time  /  is 

M       R   dt' 

where  E^  Ry  and  L  are  given  constants.  Find  the  value  of  C,  de- 
termining the  constant  of  integration  by  the  condition  that  its  initial 
value  shall  be  zero. 

Art.  2.    Geometrical  Representation. 

The  meaning  of  a  differential  equation  may  be  graphically 
illustrated  by  supposing  simultaneous  values  of  x  and  y  to  be 
the  rectangular  coordinates  of  a  variable  point.  It  is  conven- 
ient to  put/  for  the  value  of  the  ratio  dy  :  dx.  Then  /*  being 
the  moving  point  {x,  y)  and  0  denoting  the  inclination  of  its 
path  to  the  axis  of  x^  we  have 

dy 
/  =  -f-  =  tan  0. 
^       dx 

The  given  differential  equation  of  the  first  order  is  a  relation 
between  /,  x^  and  y,  and,  being  of  the  first  degree  with  respect 
to/,  determines  in  general  a  single  value  of/  for  any  assumed 
values  of  x  and^.  Suppose  in  the  first  place  that,  in  addition 
to  the  differential  equation,  we  were  given  one  pair  of  simul- 
taneous values  of  x  and  y,  that  is,  one  position  of  the  point  P, 
Now  let  P  start  from  this  fixed  initial  point  and  begin  to  move 
in  either  direction  along  the  straight  line  whose   inclination 


306 


DIFFERENTIAL    EQUATIONS. 


[Chap.  VII. 


is  determined  by  the  value  of  p  corresponding  to  the  initial 
values  of  x  and  y.  We  thus  have  a  moving  point  satisfying 
the  given  differential  equation.  As  the  point  P  moves  the 
values  of  x  and  y  vary,  and  we  must  suppose  the  direction  of 
its  motion  to  vary  in  such  a  way  that  the  simultaneous  values 
of  X,  y,  and  p  continue  to  satisfy  the  differential  equation.  In 
that  case,  the  path  of  the  moving  point  is  said  to  satisfy  the 
differential  equation.  The  point  P  may  return  to  its  initial 
position,  thus  describing  a  closed  curve,  or  it  may  pass  to  infin- 
ity in  each  direction  from  the  initial  point  describing  an  infinite 
branch  of  a  curve.*  The  ordinary  cartesian  equation  of  the 
path  o(  Pis  SL  particular  integral  of  the  differential  equation. 

If  no  pair  of  associated  values  of  x  and  y  be  known,  P  may 
be  assumed  to  start  from  any  initial  point,  so  that  there  is  an 
unhmited  number  of  curves  representing  particular  integrals 
of  the  equation.  These  form  a  "system  of  curves,"  and  the 
complete  integral  is  the  equation  of  the  system  in  the  usual 
form  of  a  relation  between  x^  y,  and  an  arbitrary  '*  parameter." 
This  parameter  is  of  course  the  constant  of  integration.  It  is 
constant  for  any  one  curve  of  the  system,  and  different  values 
of  it  determine  different  members  of  the.  system  of  curves,  or 
different  particular  integrals. 

As  an  illustration,  let  us  take  equation  (4)  of  Art.  i,  which 

may  be  written 

dy  y 

dx  x' 

Denoting  by  S  the  inclination  to 
the  axis  of  x  of  the  line  joining  P 
with  the  origin,  the  equation  is 
equivalent  to  tan  0  =  —  tan  6,  and 
therefore  expresses  that  P  moves 
in  a  direction  inclined  equally  with 
OP  to  either  axis,  but  on  the  other 


*  When  -the  form  of  the  functions  M  and  N  is  unrestricted,  there  is  no 
reason  why  either  of  these  cases  should  exist,  but  they  commonly  occur  among 
5uch  differential  equations  as  admit  of  solution. 


Art.  3.]  PRIMITIVE    OF    A    DIFFERENTIAL    EQUATION.  307 

side.  Starting  from  any  position  in  the  plane,  the  point  P 
thus  moving  must  describe  a  branch  of  an  hyperbola  having 
the  two  axes  as  its  asymptotes ;  accordingly,  the  complete 
integral  xy  ^=i  C  \'$>  the  equation  of  the  system  consisting  of 
these  hyperbolas. 

Prob.  5.  Write  the  differential  equation  which  requires  P  to  move 
in  a  direction  always  perpendicular  to  OP^  ard  thence  derive  the 
equation  of  the  system  of  curves  described. 

<^-|  =  -f'-'+^'  =  ^-) 

Prob.  6.  What  is  the  system  described  when  0  is  the  comple- 
ment of  0}  (Ans.  ^'  —  y  =  C.) 

Prob.  7.  If  0  =  26^,  show  geometrically  that  the  system  described 
consists  of  circles,  and  find  the  differential  equation. 

(Ans.  2xydx  —  (pi'  —  y'')dy^ 

Art.  3.    Primitive  of  a  Differential  Equation. 

Let  us  now  suppose  an  ordinary  relation  between  x  and  y, 
which  may  be  represented  by  a  curve,  to  be  given.  By  differ- 
entiation we  may  obtain  an  equation  of  which  the  given  equa- 
tion is  of  course  a  solution  or  particular  integral.  But  by 
combining  this  with  the  given  equation  any  number  of  differ- 
ential equations  of  which  the  given  equation  is  a  solution  may 
be  found.     For  example,  from 

y  =  m{x  —  a)  (i) 

we  obtain  directly 

2ydy  =  mdx,  (2) 

of  which  equation  (i)  is  an  integral;  again,  dividing  (2)  by  (i) 
we  have 

2dy         dx 

and  of  thisequation  also  (i)  is  an  integral. 

If  in  equation  (i)  m  be  regarded  as  an  arbitrary  parameter, 
it  is  the  equation  of  a  system  of  parabolas  having  a  common 
axis  and  vertex.  The  differential  equation  (3),  which  does  not 
contain  ;«,  is  satisfied  by  every  member  of  this  system  of  curves. 


308  DIFFERENTIAL   EQUATIONS.  [Chap.  VIL 

Hence  equation  (i)  thus  regarded  is  the  complete  integral  of 
equation  (3),  as  will  be  found  by  solving  the  equation  in  which 
the  variables  are  already  separated. 

Now  equation  (3)  is  obviously  the  only  differential  equation 
independent  of  m  which  could  be  derived  from  (i)  and  (2),  since 
it  is  the  result  of  eliminating  m.  It  is  therefore  the  "  differ- 
ential equation  of  the  system  ; "  and  in  this  point  of  view  the 
integral  equation  (i)  is  said  to  be  its  "  primitive." 

Again,  if  in  equation  (i)  ^  be  regarded  as  the  .arbitrary  con- 
stant, it  is  the  equation  of  a  system  of  equal  parabolas  having 
a  common  axis.  Now  equation  (2)  which  does  not  contain  a 
is  satisfied  by  every  member  of  this  system  of  curves;  hence  it 
is  the  differential  equation  of  the  system,  and  its  primitive  is 
equation  (i)   with  a  regarded  as  the  arbitrary  constant. 

Thus,  a  primitive  is  an  equation  containing  as  well  as  x  and 
y  an  arbitrary  constant,  which  we  may  denote  by  (7,  and  the 
corresponding  differential  equation  is  a  relation  between  x,  y^ 
and  /,  which  is  found  by  differentiation,  and  elimination  of  C  if 
necessary.  This  is  therefore  also  a  method  of  verifying  the  com- 
plete integral  of  a  given  differential  equation.  For  example,  in 
verifying  the  complete  integral  (7)  in  Art;  I  we  obtain  by  differ- 
entiation I  =  —  Cp.  If  we  use  this  to  eliminate  C  from  equa- 
tion (7)  the  result  is  equation  (6);  whereas  the  process  before 
employed  was  equivalent  to  eliminating  /  from  equation  (6), 
thereby  reproducing  equation  (7). 

Prob.  8.  Write  the  equation  of  the  system  of  circles  in  Prob.  7, 
Art.  2,  and  derive  the  differential  equation  from  it  as  a  primitive. 

Prob.  9.  Write  the  equation  of  the  system  of  circles  passing 
through  the  points  (o,  3)  and  (o,  —  ^),  and  derive  from  it  the  differ- 
ential equation  of  the  system. 

Art.  4.    Exact  Differential  Equations. 

In  Art.  I  the  case  is  mentioned  in  which  Mdx  -\-  Ndy  is  an 
"  exact  differential,"  that  is,  the  differential  of  a  function  of  x 
and  y.     Let  u  denote  this  function;  then 

du  =  Mdx -{- Ndy,  ^i) 


Art.  4.]  exact  differential  equations.  309 

and  in  the  notation  of  partial  derivatives 

■dx  ■dy 

Then,  since  by  a  theorem  of  partial  derivatives 


dydx      dxdy 
^  =  ^.  (2) 

This  condition  must  therefore  be  fulfilled  by  M  and  N  in 
order  that  equation  (i)  may  be  possible.  When  it  is  fulfilled 
Mdx  -\-  Ndy  =  o  is  said  to  be  an  "  exact  differential  equation," 
and  its  complete  integral  is 

u=^C.  ■--  (3) 

For  example,  given  the  equation 

xi^x  +  2y)dx  4-  (;r'  —  f)dy  =  o, 

M  =  x{x  -\-  2y),  N=  x^  —  y\  — —  =  2x,  and  — —  =  2x ;  the 

condition  (2)  is  fulfilled,  and  the  equation  is  exact.  To  find  the 
function  u^  we  ma}^  integrate  Mdx,  treatingj^  as  a  constant;  thus, 

in  which  the  constant  of  integration  F  may  be  a  function  of  y. 
The  result  of  differentiating  this  is 

x*dx  -f-  2xy  dx  +  x'^dy  =  dV, 

which  should  be  identical  with  the  given  equation  ;  therefore, 
^Y  =  y  dy,  whence  V  =z  ^y -\- C,  and  substituting,  the  com- 
plete integral  may  be  written 

The  result  is  more  readily  obtained  if  we  notice  that  all 
terms  containing  x  and  dx  only,  or  y  and  dy  only,  are  exact 
differentials ;  hence  it  is  only  necessary  to  examine  the  terms 
containing  both  x  and  y.  In  the  present  case,  these  are 
2xy  dx  -\-  x^dy,  which  obviously  form  the  differential  of  xy  ; 
whence,  integrating  and  multiplying  by  3,  we  obtain  the  result 
above. 

The  complete  integral  of  any  equation,  in  whatever  way  it 


310  DIFFERENTIAL    EQUATIONS.  [ChaP.  VIL 

was  found,  can  be  put  in  the  form  u  =  C,  by  solving  for  C. 
Hence  an  exact  differential  equation  du  =o  can  be  obtained, 
which  must  be  equivalent  to  the  given  equation 

Mdx  +  iV^/  =  o,  (4) 

here  supposed  not  to  be  exact.  The  exact  equation  du  =  o 
must  therefore  be  of  the  form 

M(MdA:  +  Ndy)  =  o,  (5) 

where  /<  is  a  factor  containing  at  least  one  of  the  variables  x 
and_)/.  Such  a  factor  is  called  an  "  integrating  factor"  of  the 
given  equation.  For  example,  the  result  of  differentiating 
equation  (7),  Art.  i,  when  put  in  the  form  u  =  Cj  is 

so  that  (i  —  y)"'  is  an  integrating  factor  of  equation  (6).  It 
is  to  be  noticed  that  the  factor  by  which  we  separated  the 
variables,  namely,  (i  —  f)~'(i  —  x)~\  is  also  an  integrating 
factor. 

It  follows  that  if  an  integrating  factor  can  be  discovered, 
the  given  differential  equation  can  at  once  be  solved.*  Such 
a  factor  is  sometimes  suggested  by  the  form  of  the  equation. 

Thus,  given  (jj/  —  x)df-\-ydx  =  o, 

the  terms  j/dx  —  xdy,  which  contain  both  x  and  y,  are  not  ex- 
act, but  become  so  when  divided  by  either  x'^  or  j^';  and  be- 
cause the  remaining  term  contains  y  only,  y~^  is  an  integrating 
factor  of  the  whole  expression.     The  resulting  integral  is 

^ogy  +  ^  =  C, 

Prob.  10.  Show  from  the  integral  equation  in  Prob.  9,  Art.  3,  that 
x'"^  is  an  integrating  factor  of  the  differential  equation. 

Prob.  II.  Solve  the  equation  x(x'^  +  ^y^)dx  -{-y(y^  +  3^^)^y  =  o- 

(Ans.  X*  +  6^y  +/  =  c.) 

*  Since  //J/ and  uJV  in  the  exact  equation  (5)  must  satisfy  the  condition  (2), 
we  have  a  partial  differential  equation  for  ju ;  but  as  a  general  method  of  findings 
jn  this  simply  comes  back  to  the  solution  of  the  original  equation. 


Art.  5.]  homogeneous  equation.  311 

X  dy  —  y  dx 
Prob.  1 2.  Solve  the  equation  y  dy-\-xdx-\ ^         ,  ■. 

(Ans.  ^l±ZVtan-'^=..) 

2  X 

Prob.  13.  If  «  =  ^  is  a  form  of  the  complete  integral  and  ^  the 
corresponding  integrating  factor,  show  that  ^f{u)  is  the  general 
expression  for  the  integrating  factors. 

Prob.  14.  Show  that  the  expression  x'^y^{mydx  +  nxdy)  has  the 
integrating  factor  ^*'«-'-a^*«-'-^l  and  by  means  of  such  a  factor 
solve  the  equation  ^(y  +  2x^)dx  -\-  x(^x^  —  2y^)dy  =  o. 

(Ans.   2x*y  —  y  =  cx^.) 

Prob.  15.  Solve  {x^  -\-  y^)dx  —  2xy  dy  =  o.     (Ans.  x^  —  y^  =■  cx^ 

Art.  5.    Homogeneous  Equation. 

The  differential  equation  Mdx  -\-  Ndy  =  o   is  said  to   be 

homogeneous  v^hen  M  and  N  are  homogeneous  functions  of 

X  and  y  of  the  same  degree  ;  or,  what  is  the  same  thing,  v^^hen 

dy  y 

-J-  is  expressible  as  a  function  of  — .     If  in  such  an  equation 

the  variables  are  changed  from  x  and  y  to  x  and  v,  M^here 

y 

^  =  — ;         whence    y  z^  xv     and     dy  =  xdv  +  vdx, 

the  variables  x  and   v  will  be  separable.     For  example,  the 

equation 

{x  —  2y)dx  -\-  ydy  =  o 

is    homogeneous ;    making   the   substitutions    indicated    and 
dividing  by  x^ 

(i  —  2v)dx  +  v{xdv  +  "^dx)  =  o, 

dx           vdv 
whence  h  7 ri  =  o. 

Integrating,  log  x  +  log  {v  —  \)  —  ^^— j^  =  C\ 

and  restoring^, 

The  equation  Mdx  +  Ndy  =  o  can  always  be  solved  when 


log(^_^)__— =  C. 


312  DIFFERENTIAL    EQUATIONS.  [ChAP.  VII. 

M  and  N  are  functions  of  the  first  degree,  that  is,  when  it  is 
of  the  form 

{ax  -{-  by-{-  c)dx  +  {a'x  +  b'y  +  c')dy  =  o. 
For,  assuming  x  =  x'  -\-  h,  y  =  y'  -\-  k^  it  becomes  v 

{ax'+  b'y'-\-  ah  +  bk-^r  c)dx' +{a' x' -\-  b'y'-\-  a' h  -\.b' k-\-c')dy'  =o, 
which,  by  properly  determining  h  and  k,  becomes 

{ax'  +  by')dx'  +  (a'x'  +  b'y')dy', 
a.  homogeneous  equation. 

This  method  fails  when  a  :  b  =  a' :  b\  that  is,  when  the 
equation  takes  the  form 

{ax  -\-  by  -\-  c)dx  -\-  [m{ax  +  by)  -\-  c'~\dy  =  o ; 

but  in  this  case  if  we  put  2  :=  ax  -{-  by^  and  eliminate  y^  it  will 
be  found  that  the  variables  x  and  z  can  be  separated. 

Prob.  1 6.  Show  that  a  homogeneous  differential  equation  repre- 
sents a  system  of  similar  and  similarly  situated  curves,  the  origin 
being  the  center  of  similitude,  and  hence  that  the  complete  integral 
may  be  written  in  a  form  homogeneous  in  x,  y,  and  c. 

Prob.  17.  Solve  xdy  —  ydx  —  ^{x"  -{- y^)dx  =  o. 

(Ans.  x"*  =  c'  —  2cy,) 

Prob.  18.  Solve  (^  —  Tx  -{-  'j)dx  -j-  {'jy  —  3^  +  $)dy  =  o. 

(Ans.  {y-:^^\y{y^x-iY  =  c.) 

Prob.  19.  Solve  {x^  -\-y^)dx  —  2xy  dy  =  o.     (Ans.  x^  — y  =  cx^ 

Prob.  20.  Solve  (i  +  xy)ydx  +  (i  —  xy)xdy  =  o  by  introducing 

the  new  variable  z  =  xy.  (Ans.  x  =  Cye^.) 

Prob.  21.  Solve -j^=ax-^by-\-c.     (Ans.  abx-^b^y-{-a+bc=Ce^'',) 

Art.  6.    The  Linear  Equation. 

A  differential  equation  is  said  to  be  *'  linear  "  when  (one  of 
the  variables,  say  x,  being  regarded  as  independent,)  it  is  of 
the  first  degree  with  respect  to  j/,  and  its  derivatives.  The 
linear  equation  of  the  first  order  may  therefore  be  written  in 
the  form 

J+/>=a  (I) 


Art.  6.]  THE    LINEAR    EQUATION.  313 

where  P  and  Q  are  functions  of  x  only.  Since  the  second 
member  is  a  function  of  x,  an  integrating  factor  of  the  first 
member  will  be  an  integrating  factor  of  the  equation  provided 
it  contains  x  only.     To  find  such  a  factor,  we  solve  the  equation 

%^Py  =  o,  (3) 

dy 
which  is  done  by  separating  the  variables ;  thus,  —  =  —  Pdx ; 

whence  log  y  =  c  —  j  Pdx  or 

y  =  Ce-f'"-.  (3) 

Putting  this  equation  in  the  form  u  =  Cy  the  corresponding 
exact  equation  is 

e^^^'^idy  +  Pydx)  =  o, 

whence  r  "*  is  the  integrating  factor  required.  Using  this 
factor,  the  general  solution  of  equation  (i)  is 

ef^'y  =f/'''^Qdx  +  C,  (4) 

In  a  given  example  the  integrating  factor  should  of  course 
be  simplified  in  form  if  possible.     Thus 

(i  +  x^^dy  =  (w  -|-  xy^dx 
is  a  linear  equation  for  y  \  reduced  to  the  form  (i),  it  is 

dy  X  m 

li  ~~  T+x'^  ^  I  +x'' 
from  which 

The  integrating  factor  is,  therefore. 


i/{i+xr 

whence  the  exact  equation  is 

dy  xy  dx  mdx 

4/(i+0~(i+^')'"(i+^T 


314  DIFFERENTIAL   EQUATIONS.  [Chap.  VII. 


Integrating,  there  is  found 

y  mx 


:n  +  C 


4/(1+;^")       i/(i+^0 
or 

y  =  mx  +  C  |/(l  +  x"). 

An  equation  is  sometimes   obviously  linear,  not  iox  y,  but 

for  some  function  of  y.     For  example,  the  equation 

dy  , 

-J-  -f"  tan  y  =.  X  sec  y 

when  multiplied  by  cos  y  takes  ^  form  linear  for  sin  y ;  the 
integrating  factor  is  ^,  and  the  complete  integral 

sin  J/  =  ;ir  —  I  -f-  ^^  ~* 

dy 
In  particular,  the  equation  -^  -\-  Py  =  Qy"",  which  is  known  as 

**  the  extension  of  the  linear  equation,"  is  readily  put  in  a  form 
linear  for  y^~'^ , 

dy  - 

Prob.  22.  Solve  x^—  +  (i  —  2x)y  =  x*.     (Ans.  y  =  x'^{i-{-  ce*).} 

dy 
Prob.  23.  Solve  cos  x  ~ — \-y  —  i  -(-  sm  ^  =  o. 

(Ans.  j(sec  x  +  tan  x)  =■  x  -\-  c) 

dy 
Prob.  24.  Solve  —  cos  ^  +  ^  sin  :r  =  i. 
(tx 

(Ans.  ^  =  sin  ^  +  ^  cos  x.) 
Prob.  25.  Solve  -j-  =  ^V  —  ^7.        (Ans.   -^  =  x^  -{-  i  -\-ce^*.) 

Prob.  26.  Solve  ^  = r^-^.       (Ans.  -  =  2  -/+  ^^"i^'.) 


Art.  7.    First  Order  and  Second  Degree. 

If  the  given  differential  equation  of  the  first  order,  or  re- 
lation between  x,  y,  and  /,  is  a  quadratic  for  p,  the  first  step 
in  the  solution  is  usually  to  solve  for  /.  The  resulting  value 
of  /  will  generally  involve  an  irrational  function  of  x  and  y ; 
so  that  an  equation  expressing  such  a  value  of  /,  like  some  of 
those  solved  in  the  preceding  pages,  is  not  properly  to  be  re- 


Art.  7.]  FIRST    ORDER    AND    SECOND    DEGREE.  315 

garded  as  an  equation  of  the  first  degree.  In  the  exceptional 
case  when  the  expression  whose  root  is  to  be  extracted  is  a 
perfect  square,  the  equation  is  decomposable  into  two  equa- 
tions properly  of  the  first  degree.     For  example,  the  equation 

xy(i -\- AP')  =  2p{x^ -\- f) 

y  X 

when  solved  for  p  gives  2p  =  -,  or  2p  =  -;   it   may  therefore 

be  written  in  the  form 

{2px  -  y){2py  -  x)  =  o, 
and  is  satisfied  by  putting  either 

dy       y  dy       X 

dx      2x  dx      2y 

The  integrals  of  these  equations  are 

y""  z=  ex     and     2^  —;»;''=  dT, 

and  these  form  two  entirely  distinct  solutions  of  the  given 
equation. 

As  an  illustration  of  the  general  case,  let  us  take  the  equation 

-/=/.      or      t=±%  (I) 

Separating  the  variables  and  integrating, 

i/^  ±Vy=  ±Vcy  (2) 

and  this  equation  rationalized  becomes 

(^  -  }^y  -  2C{X  -J^y)J^c'  =  0.  (3) 

There  is  thus  a  single  complete  integral  containing  one  arbi- 
trary constant  and  representing  a  single  system  of  curves; 
namely,  in  this  case,  a  system  of  parabolas  touching  each  axis 
at  the  same  distance  c  from  the  origin.  The  separate  equa- 
tions given  in  the  form  (2)  are  merely  branches  of  the  same 
parabola. 

Recurring  now  to  the  geometrical  interpretation  of  a  differ- 
ential equation,  as  given  in  Art.  2,  it  was  stated  that  an  equa- 
tion of  the  first  degree  determines,  in  general,  for  assumed 
values  of  x  and  y^  that  is,  at  a  selected  point  in  the  plane,  a 
single  value  of  /.     The  equation  was,  of  course,  then  supposed 


316  DIFFERENTIAL    EQUATIONS.  [ChAP.  VII. 

rational  in  x  and  y!^  The  only  exceptions  occur  at  points  for 
which  the  value  of  p  takes  the  indeterminate  form  ;  that  is, 
the  equation  being  Mdx  +  Ndy  =  o,  at  points  (if  any  exist) 
for  which  M=  o  and  N  =  o.  It  follows  that,  except  at  such 
points,  no  two  curves  of  the  system  representing  the  complete 
integral  intersect,  while  through  such  points  an  unlimited  num- 
ber of  the  curves  may  pass,  forming  a  "  pencil  of  curves."  f 

On  the  other  hand,  in  the  case  of  an  equation  of  the  second 
degree,  there  will  in  general  be  two  values  of  /  for  any  given 
point.  Thus  from  equation  (i)  above  we  find  for  the  point 
(4,  i),  p  =z  i  J;  there  are  therefore  two  directions  in  which  a 
point  starting  from  the  position  (4,  i)  may  move  while  satis- 
fying the  differential  equation.  The  curves  thus  described 
represent  two  of  the  particular  integrals.  If  the  same  values 
of  X  and  y  be  substituted  in  the  complete  integral  (3),  the  re- 
sult is  a  quadratic  for  c,  giving  ^  =  9  and  ^  =  i,  and  these 
determine  the  two  particular  integral  curves,  Vx  -f-  Vj^  =  3, 
and  Vx  —  Vy  =  i. 

In  like  manner  the  general  equation  of  the  second  degree, 
which  may  be  written,  in  the  form 

Lp'  +  Mp-\-N=o, 
where  Z,  M,  and  N  are  one-valued  functions  of  x  and  y,  repre- 
sents  a  system  of  curves  of  which  two  intersect  in  any  given 
point  for  which  /  is  found  to  have  two  real  values.  For  these 
points,  therefore,  the  complete  integral  should  generally  give 
two  real  values  of  c.  Accordingly  we  may  assume,  as  the 
standard  form  of  its  equation, 

Pc'  +  Qc  +  R  =  o, 

*  In  fact  /  was  supposed  to  be  a  one- valued  function  of  x  and  ^;  thus, 
/  =  sin~^jc  would  not  in  this  connection  be  regarded  as  an  equation  of  the  first 
degree. 

f  In  Prob.  6,  Art.  3,  the  integral  equation  represents  the  pencil  of  circles  pass- 
ing through  the  points  (o,  d)  and  (o,  —  3);  accordingly/  in  the  differential  equa- 
tion is  indeterminate  at  these  points.  In  some  cases,  however,  such  a  point  is 
merely  a  node  of  one  particular  integral.  Thus  in  the  illustration  given  in  Art.  2, 
J>  is  indeterminate  at  the  origin,  and  this  point  is  a  node  of  the  only  particular 
integral,  xy  =  o,  which  passes  through  it. 


Art.  8.]  singular  solutions.  317 

where  P^  Q,  and  R  are  also  one-valued  functions  of  x  and  y. 
If  there  are  points  which  make  /  imaginary  in  the  differential 
equation,  they  will  also  make  c  imaginary  in  the  integral. 

Prob.  27.  Solve  the  equation  p"  -\-  y^  =  i  and  reduce  the  inte- 
gral to  the  standard  form. 

(Ans.  {y  +  cos  x)c^  —  2c  sin  x  -\-  y  —  cos  x  =  o.) 

Prob.  28.  Solve  J/''  +  2xj>  —  y  =  o,  and  show  that  the  intersect- 
ing curves  at  any  given  point  cut  at  right  angles. 

Prob.  29.  Solve  {x''  +  i)/'  =  i.  (Ans.  cV^  —  2cxcy  =1.) 

Art.  8.    Singular  Solutions. 

A  differential  equation  not  of  the  first  degree  sometimes 
admits  of  what  is  called  a  "  singular  solution  ;  "  that  is  to  say,  a 
solution  which  is  not  included  in  the  complete  integral.  For 
suppose  that  the  system  of  curves  representing  the  complete 
integral  has  an  envelope.  Every  point  A  of  this  envelope 
is  a  point  of  contact  with  a  particular  curve  of  the  complete  in- 
tegral system  ;  therefore  a  point  moving  in  the  envelope  when 
passing  through  A  has  the  same  values  of  x^  y^  and  /  as  if  it 
were  moving  through  A  in  the  particular  integral  curve.  Hence 
such  a  point  satisfies  the  differential  equation  and  will  continue 
to  satisfy  it  as  long  as  it  moves  in  the  envelope.  The  equation, 
of  the  envelope  is  therefore  a  solution  of  the  equation. 

As  an  illustration,  let  us  take  the  system  of  straight  lines 
whose  equation  is 

J^  =  ^^  +  7,  (i) 

c 

where  c  is  the  arbitrary  parameter.     The  differential  equation 

derived  from  this  primitive  is 

y=px-\--.,  (2) 

P 
of  which  therefore  (i)  is  the  complete  integral. 

Now  the  lines  represented  by  equation  (i),  for  different 
values  of  Cy  are  the  tangents  to  the  parabola 

/  =  d^x,  (3) 


318 


DIFFERENTIAL    EQUATIONS. 


[Chap.  VII. 


A  point  moving  in  this  parabola  has  the  same  value  of/  as  if  it 

were  moving  in  one  of  the  tan- 
gents, and  accordingly  equation 
(3)  will  be  found  to  satisfy  the 
differential  equation  (2). 

It  will  be  noticed  that  for 
any  point  on  the  convex  side  of 
the  parabola  there  are  two  real 
values  of  p  ;  for  a  point  on  the 
other  side  the  values  of  /  are 
imaginary,  and  for  a  point  on 
the  curve  they  are  equal.  Thus 
its  equation  (3)  expresses  the 
relation  between  x  and  y  which  must  exist  in  order  that  (2) 
regarded  as  a  quadratic  for  p  may  have  equal  roots,  as  will  be 
seen  on  solving  that  equation. 

In  general,  writing  the  differential  equation  in  the  form 

^/  +  ^/+^  =  o,  (4) 

the  condition  of  equal  roots  is 

M'  -^LN  =  o.  (5) 

The  first  member  of  this  equation,  which  is  the  '*  discrimi- 
nant "  of  equation  (4),  frequently  admits  of  separation  into 
factors  rational  in  x  andj.  Hence,  if  there  be  a  singular  solu- 
tion, its  equation  will  be  found  by  putting  the  discriminant  of 
the  differential  equation,  or  one  of  its  factors,  equal  to  zero. 

It  does  not  follow  that  every  such  equation  represents  a  solu- 
tion of  the  differential  equation.  It  can  only  be  inferred  that 
it  is  a  locus  of  points  for  which  the  two  values  of  /  become 
equal.  Now  suppose  that  two  distinct  particular  integral 
curves  touch  each  other.  At  the  point  of  contact,  the  two 
values  of/,  usually  distinct,  become  equal.  The  locus  of  such 
points  is  called  a  "tac-locus."  Its  equation  plainly  satisfies  the 
discriminant,  but  does  not  satisfy  the  differential  equation.  An 
illustration  is  afforded  by  the  equation 


Art.  8.]  singular  solutions.  319 

of  which  the  complete  integral  is  y^  ~\-  (x  —  cY  =  a',  and  the 
discriminant,  see  equation  (5),  isy(y  —  a"")  =0. 

This  is  satisfied  by  j  =  ^,  jV  =  —  Uj  and  f  =  o,  the  first  two 
of  which  satisfy  the  differential  equation,  while  2  =^  o  does  not. 
The  complete  integral  represents  in  this  case  all  circles  of  radius 
a  with  center  on  the  axis  of  x.  Two  of  these  circles  touch  at 
every  point  of  the  axis  of  Xy  which  is  thus  a  tac-locus,  while 
jf  =  a  and  j/  =  —  a  constitute  the  envelope. 

The  discriminant  is  the  quantity  which  appears  under  the 
radical  sign  when  the  general  equation  (4)  is  solved  for  />,  and 
therefore  it  changes  sign  as  we  cross  the  envelope.  But  the 
values  of  /  remain  real  as  we  cross  the  tac-locus,  so  that  the 
discriminant  cannot  change  sign.  Accordingly  the  factor  which 
indicates  a  tac-locus  appears  with  an  even  exponent  (as  y  in 
the  example  above),  whereas  the  factor  indicating  the  singular 
solution  appears  as  a  simple  factor,  or  with  an  odd  exponent. 

A  simple  factor  of  the  discriminant,  or  one  with  an  odd  ex- 
ponent, gives  in  fact  always  the  boundary  between  a  region  of 
the  plane  in  which  /  is  real  and  one  in  which  /  is  imaginary  ; 
nevertheless  it  may  not  give  a  singular  solution.  For  the  two 
arcs  of  particular  integral  curves  which  intersect  in  a  point  on 
the  real  side  of  the  boundary  may,  as  the  point  is  brought  up 
to  the  boundary,  become  tangent  to  each  other,  but  not  to  the 
boundary  curve.  In  that  case,  since  they  cannot  cross  the 
boundary,  they  become  branches  of  the  same  particular  inte- 
gral forming  a  cusp.  A  boundary  curve  of  this  character  is 
called  a  "  cusp-locus  "  ;  the  value  of  /  for  a  point  moving  in  it 
is  of  course  different  from  the  equal  values  of/  at  the  cusp,  and 
therefore  its  equation  does  not  satisfy  the  differential  equation.* 

Prob.  30.  To  what  curve  is  the  line  y  =  mx  -\^  a  ^/{i  —  w') 
always  tangent  ?  (Ans.  y^  —  x''  =  «'.) 

Prob.  31.  Show  that  the  discriminant  of  a  decomposable  differ- 

*  Since  there  is  no  reason  why  the  values  of/  referred  to  should  be  identical, 
we  conclude  that  the  equation  Z/^  +  Mp  +  ^=  o  tias  not  in  general  a  singular 
solution,  its  discriminant  representing  a  cusp-locus  except  when  a  certain  con- 
dition is  fulfilled. 


330  DIFFERENTIAL    EQUATIONS.  [ChAP.  VII. 

ential  equation  cannot  be  negative.  Interpret  the  result  of  equating 
it  to  zero  in  the  illustrative  example  at  the  beginning  of  Art.  7. 

Prob.  32.  Show  that  the  singular  solutions  of  a  homogeneous  dif- 
ferential equation  represent  straight  lines  passing  through  the  origin. 

Prob.  33.  Solve  the  equation  xp^  —  2yp  -\-  ax  =  o. 

( Ans.  x^  —  2cy  -{-  ac"^  =  o  ;  singular  solution  y  =  ax^,) 

Prob.  34.  Show  that  the  equation  /'  +  2xp  —  y=  o  has  no  sin- 
gular solution,  but  has  a  cusp-locus,  and  that  the  tangent  at  every 
cusp  passes  through  the  origin. 

Art.  9.    Singular  Solution  from  Complete  Integral. 

When  the  complete  integral  of  a  differential  equation  of 
the  second  degree  has  been  found  in  the  standard  form 

Pc'+Qc  +  R  =  o  (l> 

(see  the  end  of  Art.  7),  the  substitution  of  special  values  of  x 
and  y  in  the  functions  P,  Q,  and  R  gives  a  quadratic  for  c  whose 
roots  determine  the  tvjo  particular  curves  of  the  system  which 
pass  through  a  given  point.  If  there  is  a  singular  solution^ 
that  is,  if  the  system  of  curves  has  an  envelope,  the  two 
curves  which  usually  intersect  become  identical  when  the  given 
point  is  moved  up  to  the  envelope.  Every  point  on  the  en- 
velope therefore  satisfies  the  condition  of  equal  roots  for  equa- 
tion (i),  which  is 

Q-ArPR  =  o;  (2) 

and,  reasoning  exactly  as  in  Art.  8,  we  infer  that  the  equation 
of  the  singular  solution  will  be  found  by  equating  to  zero  the 
discriminant  of  the  equation  in  c  or  one  of  its  factors.  Thus 
the  discriminant  of  equation  (i),  Art.  8,  or  "^-discriminant,"  is 
the  same  as  the  "/-discriminant,"  namely,  _;'''  —  /[ax,  which 
equated  to  zero  is  the  equation  of  the  envelope  of  the  system  of 
straight  lines. 

But,  as  in  the  case  of  the  /-discriminant,  it  must  not  be 
inferred  that  every  factor  gives  a  singular  solution.  For  ex- 
ample, suppose  a  squared  factor  appears  in  the  r-discriminant. 
The  locus  on  which  this  factor  vanishes  is  not  a  curve  in  cross- 
ing which  c  and/  become  imaginary.     At  any  point  of  it  there 


Art.  9.]       SINGULAR  SOLUTION  FROM  COMPLETE  INTEGRAL.  331 

will  be  two  distinct  values  of  /,  corresponding  to  arcs  of  par- 
ticular integral  curves  passing  through  that  point ;  but.  since 
there  is  but  one  value  of  c,  these  arcs  belong  to  the  same  par- 
ticular integral,  hence  the  point  is  a  double  point  or  node. 
The  locus  is  therefore  called  a  "  node-locus."  The  factor  repre- 
senting it  does  not  appear  in  the  /-discriminant,  just  as  that 
representing  a  tac-locus  does  not  appear  in  the  ^-discriminant. 

Again,  at  any  point  of  a  cusp-locus,  as  shown  at  the  end  of 
Art.  8,  the  two  branches  of  particular  integrals  become  arcs  of 
the  same  particular  integral ;  the  values  of  c  become  equal,  so 
that  a  cusp-locus  also  makes  the  ^-discriminant  vanish. 

The  conclusions  established  above  obviously  apply  also  to 
equations  of  a  degree  higher  than  the  second.  In  the  case  of 
the  ^-equation  the  general  method  of  obtaining  the  condition 
for  equal  roots,  which  is  to  eliminate  c  between  the  original  and 
the  derived  equation,  is  the  same  as  the  process  of  finding  the 
envelope  or  "  locus  of  the  ultimate  intersections  "  of  a  system 
of  curves  in  which  c  is  the  arbitrary  parameter. 

Now  suppose  the  system  of  curves  to  have  for  all  values  of 
c*  a  double  point,  it  is  obvious  that  among  the  intersections 
of  two  neighboring  curves  there  are  two  in  the  neighborhood 
of  the  nodes,  and  that  ultimately  they  coincide  with  the  node, 
which  accounts  for  the  node-locus  appearing  twice  in  the  dis- 
criminant or  locus  of  ultimate  intersections.     In  like  manner, 

*  It  is  noticed  in  the  second  foot-note  to  Art.  7  that  for  an  equation  of  the 
first  degree  p  takes  the  indeterminate  form,  not  only  at  a  point  through  which  all 
curves  of  the  system  pass  (where  the  value  of  c  would  also  be  found  indeter- 
minate), but  at  a  node  of  a  particular  integral.  So  also  when  the  equation  is  of 
the  «th  degree,  if  there  is  a  node  for  a  particular  value  of  c,  the  n  values  of  c  at 
the  point  (which  is  not  on  a  node-locus  where  two  values  of  c  are  equal)  deter- 
mine n  -f- 1  arcs  of  particular  integrals  passing  through  the  point ;  and  there- 
fore there  are  n-\-  1  distinct  values  of  p  at  the  point,  which  can  only  happen 
when/  takes  the  indeterminate  form,  that  is  to  say,  when  all  the  coefficients  of 
the /-equation  Cwhich  is  of  the  «th  degree^  vanish.  See  Cayley  on  Singular  So- 
lutions- in  the  "Messenger  of  Mathematics.  New  Series,  Vol.  II,  p.  10  (Collected 
Mathematical  Works,  Vol.  VIII,  p.  529).  The  present  t-neory  of  Singular  Solu- 
tions was  established  by  Cayley  in  this  paper  and  its  continuation,  Vol.  VI,  p.  23. 
Sec  also  a  paper  by  Dr.  Glaisher,  Vol.  XII,  p.  i. 


322  DIFFERENTIAL    EQUATIONS.  TChaP.  Vll. 

if  there  is  a  cusp  for  all  values  of  c,  there  are  three  intersections 
of  neighboring  curves  (all  of  which  may  be  real)  which  ulti- 
mately coincide  with  the  cusp;  therefore  a  cusp-locus  will 
appear  as  a  cubed  factor  in  the  discriminant."'^ 

Prob.  35.  Show  that  the  singular  solutions  of  a  homogeneous 
equation  must  be  straight  lines  passing  through  the  origin. 

Prob.  36.  Solve  3^^*/  —  2xyp  +  ^'  —  ^'^  =  o,  and  show  that  there 
is  a  singular  solution  and  a  tac-locus. 

Prob.  37.  Solve  vp^  +  2Xp  —  ^  =  o,  and  show  that  there  is  an 
imaginary  singular  solution.  (Ans.  y^  =  2cx  +  ^r'.) 

Prob.  38.  Show  that  the  equation  (i  —  x^^f  =  i  —  y  represents 
a  system  of  conies  touching  the  four  sides  of  a  square. 

Prob.  39.  Solve  yp"  —  ^xp  -\-y  =  o  ;  examine  and  interpret  both 
discriminants.  (Ans.  c"  +  2cx{2^^  —  Zx^)  —  ^x^y''  -j-y  =  o.) 


Art.  10.    Solution  by  Differentiation. 

The  result  of  differentiating  a  given  differential  equation  of 
the  first  order  is  an  equation  of  the  second  order,  that  is,  it 

d'^y 
contains  the  derivative  -7-^ ;  but,  if  it  does  not  contain  y  ex- 
plicitly, it  may  be  regarded  as  an  equatian  of  the  first  order  for 
the  variables  x  and/.  If  the  integral  of  such  an  equation  can 
be  obtained  it  will  be  a  relation  between  x,  p,  and  a  constant 
of  integration  ^,  by  means  of  which  /  can  be  eliminated  from 
the  original  equation,  thus  giving  the  relation  between  x,  j, 
and  c  which  constitutes  the  complete  integral.  For  example, 
the  equation 

J +2^/  =  ^'+/.  (I) 

*  The  discriminant  of  Pc*^  -^^  Qc  -^  Rz=z  o  represents  in  general  an  envelope, 
no  further  condition  requiring  to  be  fulfilled  as  in  the  case  of  the  discriminant 
of  Lp^  -j-  Mp  -f  iV  =  o.  Compare  the  foot-note  to  Art.  8.  Therefore  where 
there  is  an  integral  of  this  form  there  is  generally  a  singular  solution,  although 
Lp^  ■\-  Mp -\-N  —o  has  not  in  general  a  singular  solution.  We  conclude,  there- 
fore, that  this  equation  (in  which  Z,  M,  and  N  are  one-valued  functions  of  x 
and  y)  has  not  in  general  an  integral  of  the  above  form  in  which  P,  Q,  and  P 
are  one-valued  functions  of  x  and  y.  Cayley,  Messenger  of  Mathematics,  New 
Series,  Vol.  VI,  p.  23. 


Art.  10.]  SOLUTION  by  differentiation.  323 

when  solved  forj^,  becomes 

/  =  ^  +  <?;  (2) 

whence  by  differentiation 

The  variables  can  be  separated  in  this  equation,  and  its  inte- 
gral is 

Substituting  in  equation  (2),  we  find 

which  is  the  complete  integral  of  equation  (i). 

This  method  sometimes  succeeds  with  equations  of  a  higher 
degree  when  the  solution  with  respect  to  p  is  impossible  or 
leads  to  a  form  which  cannot  be  integrated.  A  differential 
equation  between  p  and  one  of  the  two  variables  will  be  ob- 
tained by  direct  integration  when  only  one  of  the  variables  is 
explicitly  present  in  the  equation,  and  also  when  the  equation 
is  of  the  first  degree  with  respect  to  x  and  y.  In  the  latter 
case  after  dividing  by  the  coefficient  of  jj/,  the  result  of  differ- 
entiation will  be  a  linear  equation  for  4r  as  a  function  of  /»,  so 
that  an  expression  for  x  in  terms  of  p  can  be  found,  and  then 
by  substitution  in  the  given  equation  an  expression  for  y  in 
terms  of  p.  Hence,  in  this  case,  any  number  of  simultaneous 
values  of  x  and  /  can  be  found,  although  the  elimination  of  p 
may  be  impracticable. 

In  particular,  a  homogeneous  equation  which  cannot  be 
solved  for  p  may  be  soluble  for  the  ratio  y  \  x,  so  as  to  assume 
the  form  y  =  xip(f).     The  result  of  differentiation  is 

di) 
p  =  ct>(p)  +  <t>'{p)£. 

in  which  the  variables  x  and  p  can  be  separated. 
Another  special  case  is  of  the  form 

y-=^P^+AP\  (I) 


324  DIFFERENTIAL   EQUATIONS.  [ChaP-  VII. 

which  is  known  as  Clairaut's  equation.     The  result  of  differ- 
entiation is 

which  implies  either 

^+/V)=o,  or         J=:0. 

The  elimination  of  /  from  equation  (i)  by  means  of  the 
first  of  these  equations  *  gives  a  solution  containing  no  arbi- 
trary constant,  that  is,  a  singular  solution.  The  second  is  a 
differential  equation  for  p ;  its  integral  is  /  =  ^,  which  in 
equation  (i)  gives  the  complete  integral 

y^cx^Ac).  (2) 

This  complete  integral  represents  a  system  of  straight  lines, 
the  singular  solution  representing  the  curve  to  which  they  are 
.all  tangent.     An  example  has  already  been  given  in  Art.  8. 

A  differential  equation  is  sometimes  reducible  to  Clairaut's 
form  by  means  of  a  more  or  less  obvious  transformation  of  the 
variables.  It  may  be  noticed  in  particular  that  an  equation  of 
the  form 

is  sometimes  so  reducible  by  transformation  to  the  independent 
variable  z,  where  x  =  z""  \  and  an  equation  of  the  form 

y^nxp-\-  <p{y,p\ 
by  transformation  to  the  new  dependent  variable  v  =  y"^.     A 
double   transformation    of   the   form  indicated    may   succeed 
when  the  last  term  is  a  function  of  both  x  and  y  as  well  as  of/. 

Prob.  40.  Solve  the  equation  3);  =  2/"  +  3/^;  find  a  singular 
solution  and  a  cusp-locus.         (Ans.  {x -\- y  -^-  c  —  ^Y  =  ^(•^+^)'0 

Prob.  41.  Solve  2y  —xp  -\-  — ,  and  find  a  cusp-locus. 

(Ans.  «V'  —  i2acxy  -\-  2>cy^  —  \2x^y^  +  ^dax*  —  o.) 

*  The  equation  is  in  fact  the  same  that  arises  in  the  general  method  for  the 
condition  of  equal  roots.     See  Art.  9. 


Art.  11.]  GEOMETRIC    APPLICATIONS  ;     TRAJECTORIES.  335 

Prob.  42.   Solve  (x^  —  a'^)j>^  —  2xyp  -f  y  —  ^'  =  o. 

(Ans.  The  circle  ^'  -f-/  =  o,  and  its  tangents.) 

Prob.  43.  Solve  y  =  -^  x/> -\-  x^p", 

(Ans.  c^x  -\-  c  —  xy  =  o,  and  i  +  ^x'y  =  o.) 

Prob.  44.   Solve/  —  4xyp  +  Sy'  =  o. 

(Ans.  y  =  c{x  ~  cY  ;  2yy  =  4^'  and  jj'  =  o  are  singula! 
solutions;  J'  =  o  is  also  a  particular  integral.) 

Prob.  45.  Solve  x\y  —  px)  =  yp\  (Ans.  /  =  ex""  +  c\) 

Art.  11.    Geometric  Applications  ;  Trajectories. 

Every  property  of  a  curve  which  involves  the  direction  of 
its  tangents  admits  of  statement  in  the  form  of  a  differential 
equation.  The  solution  of  such  an  equation  therefore  deter- 
mines the  curve  having  the  given  property.  Thus,  let  it  be 
required  to  determine  the  curve  in  which  the  angle  between 
the  radius  vector  and  the  tangent  is  n  times*  the  vectorial 
angle.  Using  the  expression  for  the  trigonometric  tangent  of 
that  angle,  the  expression  of  the  property  in  polar  coordi- 
nates is 

— —  =  tan  nd, 
dr 

Separating    the   variables   and    integrating,   the   complete 

integral  is 

r**  =  ^**  sin  nd. 

The  mode  in  which  the  constant  of  integration  enters  here 
shows  that  the  property  in  question  is  shared  by  all  the  mem- 
bers of  a  system  of  similar  curves. 

The  solution  of  a  question  of  this  nature  will  thus  in  gen- 
eral be  a  system  of  curves,  the  complete  integral  of  a  differential 
equation,  but  it  may  be  a  singular  solution.  Thus,  if  we  ex- 
press the  property  that  the  sum  of  the  intercepts  on  the  axes 
made  by  the  tangent  to  a  curve  is  equal  to  the  constant  a,  the 
straight  lines  making  such  intercepts  will  themselves  consti- 
tute the  complete  integral  system,  and  the  curve  required  is 
the  singular  solution,  which,  in  accordance  with  Art.  8,  is  the 


326  DIFFERENTIAL    EQUATIONS.  [ChaP.  VIL 

envelope  of  these  lines.     The  result  in  this  case  will  be  found 
to  be  the  parabola  Vx  +  Vy  =  Va. 

An  important  appHcation  is  the  determination  of  the 
"orthogonal  trajectories"  of  a  given  system  of  curves,  that  is 
to  say,  the  curves  which  cut  at  right  angles  every^^curve  of  the 
given  system.  The  differential  equation  of  the  trajectory  is 
readily  derived  from  that  of  the  given  system  ;  for  at  every 
point  of  the  trajectory  the  value  of/  is  the  negative  reciprocal 
of  its  value  in  the  given  differential  equation.  We  have  there- 
fore only  to  substitute  —/"'for/  to  obtain  the  differential 
equation  of  the  trajectory.  For  example,  let  it  be  required  to 
determine  the  orthogonal  trajectories  of  the  system  of  pa- 
rabolas 

having  a  common  axis  and  vertex.     The  differential  equation 
of  the  system  found  by  eHminating  a  is 

2  xdy  =  y  dx. 

Putting  —  —  in  place  of  -j-,  the  differential  equation  of 

the  system  of  trajectories  is 

2xdx  -\-  ydy  ^=  Of 
whence,  integrating, 

2x'-^/=^c\ 

The  trajectories  are  therefore  a  system  of  similar  ellipses 
with  axes  coinciding  with  the  coordinate  axes. 

Prob.  46.  Show  that  when  the  differential  equation  of  a  system, 
is  of  the  second  degree,  its  discriminant  and  that  of  its  trajectory 
system  will  be  identicar;  but  if  it  represents  a  singular  solution  in 
one  system,  it  will  constitute  a  cusp  locus  of  the  other. 

Prob.  47.  Determine  the  curve  whose  subtangent  is  constant  and 
equal  to  «.  (Ans.  ce*=y.) 

Prob.  48.  Show  that  the  orthogonal  trajectories  of  the  curves 
r^=c''  sinnO  are  the  same  system  turned  through  the  angle  —  about 
the  pole.     Examine  the  cases  n  =  i,  «  =  2,  and  n  =  i. 

Prob.  49.   Show  that  the  orthogonal  trajectories  of  a  system  of 


Art.  12.J         simultaneous  differential  equations.  3'27 

circles  passing  through  two  given  points  is  another  system  of  circles 

having  a  common  radical  axis. 

Prob.  50.   Determine    the    curve    such    that    the    area   inclosed 

by  any  two  ordinates,  the   curve    and    the    axis   of  x,   is  equal  to 

the  product  of  the    arc  and  the  constant    line  a.     Interpret   the 

singular  solution.  -       -- 

(Ans.  The  catenary  ji;  =  ia(e''—e  ").) 

Prob.  51.  Show  that  a  system  of  confocal  conies  is  self-orthog- 
onal. 

Art.  12.    Simultaneous  Differential  Equations. 

A  system  of  n  equations  between  n  -{-  1  variables  and  their 
differentials  is  a  "determinate"  differential  system,  because  it 
serves  to  determine  the  n  ratios  of  the  differentials  ;  so  that, 
taking  any  one  of  the  variables  as  independent,  the  others  vary 
in  a  determinate  manner,  and  may  be  regarded  as  functions  of 
the  single  independent  variable.  Denoting  the  variables  by  x, 
yy  Zy  etc.,  the  system  may  be  written  in  the  symmetrical  form 

dx  _dy  _dz  _ 

X~~Y~Z  ' 

where  X,  Y,  Z .  .  .  may  be  any  functions  of  the  variables. 

If  any  one  of  the  several  equations  involving  two  differen- 
tials contains  only  the  two  corresponding  variables,  it  is  an 
ordinary  differential  equation  ;  and  its  integral,  giving  a  re- 
lation between  these  two  variables,  may  enable  us  by  elimina- 
tion to  obtain  another  equation  containing  two  variables  only, 
and  so  on  until  n  integral  equations  have  been  obtained. 
Given,  for  example,  the  system 

dx  _dy  _  dz  ,  . 

X  ~    z   ~  y' 
The  relation  between  dy  and  dz  above  contains  the  varia- 
bles j  and  z  only,  and  its  integral  is 

y  —  z^  =  a.  (2> 

Employing  this  to  eliminate  z  from  the  relation  between 
dx  and  dy  it  becomes 

dx  _         dy 


328  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

of  which  the  integral  is 

/+  ^{f^a)  =  bx.  (3) 

The  integral  equations  (2)  and  (3),  involving  two  constants 
of  integration,  constitute  the  complete  solution.  It  is  in  like 
manner  obvious  that  the  complete  solution  of  a  system  of  n 
equations  should  contain  n  arbitrary  constants. 

Confining  ourselves  now  to  the  case  of  three  variables,  an 
extension  of  the  geometrical  interpretation  given  in  Art.  2 
presents  itself.  Let  x,  y,  and  z  be  rectangular  coordinates  of 
P  referred  to  three  planes.  Then,  if  P  starts  from  any  given 
position  A,  the  given  system  of  equations,  determining  the 
ratios  dx:  dy\  dz,  determines  the  direction  in  space  in  which  P 
moves.  As  P  moves,  the  ratios  of  the  differentials  (as  deter- 
mined by  the  given  equations)  will  vary,  and  if  we  suppose  P 
to  move  in  such  a  way  as  to  continue  to  satisfy  the  differential 
equations,  it  will  describe  in  general  a  curve  of  double  curva- 
ture which  will  represent  a  particular  solution.  The  complete 
solution  is  represented  by  the  system  of  lines  which  may  be 
thus  obtained  by  varying  the  position  of  the  initial  point  A. 
This  system  is  a  "  doubly  infinite  "  one  ;  for  the  two  relations 
betw^een  x,  y,  and  z  which  define  it  analytically  must  contain 
two  arbitrary  parameters,  by  properly  determining  which  we 
can  make  the  line  pass  through  any  assumed  initial  point. ^ 

Each  of  the  relations  between  x,  y  and  z,  or  integral  equa- 
tions, represents  by  itself  a  surface,  the  intersection  of  the  two 
surfaces  being  a  particular  line  of  the  doubly  infinite  system. 
An  equation  like  (2)  in  the  example  above,  which  contains  only 
one  of  the  constants  of  integration,  is  called  an  integral  of  the 
differential  system,  in  contradistinction  to  an   "  integral  equa- 

*  It  is  assumed  in  the  explanation  that  X,  V,  and  Z  are  one-valued  functions 
of  x,y,  and  z.  There  is  then  but  one  direction  in  which  P  can  move  when 
passing  a  given  point,  and  the  system  is  a  non-intersecting  system  of  lines.  But 
if  this  is  not  the  case,  as  for  example  when  one  of  the  equations  giving  the  ratio 
of  the  differentials  is  of  higher  degree,  the  lines  may  form  an  intersecting  sys- 
tem, and  there  would  be  a  theory  of  singular  solutions,  into  which  we  do  not 
here  enter. 


Art.  12.]        simultaneous  differential  equations.  329 

tion  "  like  (3),  which  contains  both  constants.  An  integral 
represents  a  surface  which  contains  a  singly  infinite  system  of 
lines  representing  particular  solutions  selected  from  the  doubly 
infinite  system.  Thus  equation  (2)  above  gives  a  surface  on 
which  lie  all  those  lines  for  which  a  has  a  given  value,  while  b 
may  have  any  value  whatever  ;  in  other  words,  a  surface  which 
passes  through  an  infinite  number  of  the  particular  solution 
lines. 

The  integral  of  the  system  which  corresponds  to  the  con- 
stant b  might  be  found  by  eliminating  a  between  equations  (2) 
and  (3).  It  might  also  be  derived  directly  from  equation  (i)  ; 
thus  we  may  write 

dx  _dy  _  dz  __   dy  -\-  dz  __dti 
X  ~~   z  ~  y  ~     y  -\-  ^     ~  u' 

in  which  a  new  variable  u  =  y  -\-  z  is  introduced.  The  rela- 
tion between  dx  and  du  now  contains  but  two  variables,  and 

its  integral, 

y  +  ^  =  bx,  (4) 

is  the  required  integral  of  the  system  ;  and  this,  together  with 
the  integral  (2),  presents  the  solution  of  equations  (i)  in  its 
•standard  form.  The  form  of  the  two  integrals  shows  that  in 
this  case  the  doubly  infinite  system  of  lines  consists  of  hyper- 
bolas, namely,  the  sections  of  the  system  of  hyperbolic  cylinders 
represented  by  (2)  made  by  the  system  of  planes  represented 

by  (4). 

A  system  of  equations  of  which  the  members  possess  a  cer- 
tain symmetry  may  sometimes  be  solved  in  the  following 
manner.     Since 

dx dy  _^dz  \dx  A-  j^dy  -\-  vdz 

if  we  take  multipliers  A,  /^,  v  such  that 

\X+fxY+yZ=o, 
we  shall  have  Tidx  -|-  f^dy  +  vdz  =  o. 

If  the  expression  in  the  first  member  is  an  exact  differential, 


330  DIFFERENTIAL    EQUATIONS.  [Chap.  VIL 

direct  integration  gives  an  integral  of  the  given  system.  For 
example,  let  the  given  equations  be 

dx       _       dy      _        dz 
niz  —  ny       nx  —  Iz       ly  —  mx  * 

/^  m  and  n  form  such  a  set  of  multipliers,  and  so  also  do  Xy  y 
and  z.     Hence  we  have 

Idx  -)-  mdy  -f-  ndz  =  o, 

and  also  xdx  -{-ydy-\-  zdz  =  o. 

Each  of  these  is  a^  <5xact  equation,  and  their  integrals 

/x  -\-  my  -\-  nz  =  a 

and  x'-{-y  +  z'=d' 

constitute  the  complete  solution.  The  doubly  infinite  system; 
of  lines  consists  in  this  case  of  circles  which  have  a  common 
axis,  namely,  the  Hne  passing  through  the  origin  and  whose 
direction  cosines  are  proportional  to  /,  m,  and  n. 

dx                 dy           dz 
Prob.  i,2.  Solve  the   equations  -^ ^ ;,  =  — ^-- = ,  and 

X  —  y  —  z         2xy        2XZ 

interpret  the  result  geometrically.      (Ans.  y=aZy  x'^-\-y^  -\-z^=bz.) 
dx  dy  dz 


Prob.  53.   Solve 


Prob.  54.  Solve 


y  -\-  z      z  -\-  X       X  -\-y 

(Ans.^(x+;-  +  .)=-^  =  ^4-J 

dx       _         dy         __        dz 
{b  —  c)yz  ~  {c  —  d)zx   "^  {a  —  b)xy' 
(Ans.  x'+y+z'  =  A,ax'4- bf  +  cz;''  =  -5.) 


Art.  13.    Equations  of  the  Second  Order. 

A  relation  between  two  variables  and  the  successive  deriva-^ 
tives  of  one  of  them  with  respect  to  the  other  as  independent 
variable  is  called  a  differential  equation  of  the  order  indicated 
by  the  highest  derivative  that  occurs.     For  example, 

is  an  equation  of  the  second  order,  in  which  x  is  the  independent 


Art.  13.]  EQUATIONS    OF    THE    SECOND    ORDER.  331 

variable.  Denoting  as  heretofore  the  first  derivative  by/,  this 
equation  may  be  written 

and  this,  in  connection  with 

which  defines  /,  forms  a  pair  of  equations  of  the  first  order,, 
connecting  the  variables  x,  y,  and  /.  Thus  any  equation  of  the 
second  order  is  equivalent  to  a  pair  of  simultaneous  equations 
of  the  first  order. 

When,  as  in  this  example,  the  given  equation  does  not  con- 
tain j  explicitly,  the  first  of  the  pair  of  equations  involves  only 
the  two  variables  x  and  /  ;  and  it  is  further  to  be  noticed  that, 
when  the  derivatives  occur  only  in  the  first  degree,  it  is  a  linear 
equation  for/.     Integrating  equation  (i)  as  such,  we  find 

^  =  --+W+-^Y  (3)- 

and  then  using  this  value  of/  in  equation  (2),  its  integral  is 

y  =  c^-mx-{-  c,  log  [^  +  v(i  +  ^')]>  (4) 

in  which,  as  in  every  case  of  two  simultaneous  equations  of  the 
first  order,  we  have  introduced  two  constants  of  integration. 

An  equation  of  the  first  order  is  readily  obtained  also 
when  the  independent  variable  is  not  explicitly  contained  in 
the  equation.     The  general  equation  of  rectilinear  motion  in 

dynamics  affords  an  illustration.     This  equation  is  —  r=/(5),. 

where  s  denotes  the  distance  measured  from  a  fixed   center  of 

force  upon  the  line  of  motion.     It  may  be  written  -r-  =/(j),  in 

at 

ds 
connection  with  —  =  v,  which  defines  the  velocity.     Eliminat- 
ing  dt   from    these    equations,   we    have   vdv  ^  f(s)ds,   whose 
integral  is  -J-z^' =    I  f{s)ds -\- c,  the  "equation  of  energy"  for 
the  unit  mass.    The  substitution  of  the  value  found  for  if  in  the 


332  DIFFERENTIAL   EQUATIONS.  [ChAP.  VII. 

second  equation  gives  an  equation  from  which  t  is  found  in 
terms  of  s  by  direct  integration. 

The  result  of  the  first  integration,  such  as  equation  (3)  above, 
is  called  a  "  first  integral "  of  the  given  equation  of  the  second 
order ;  it  contains  one  constant  of  integration,  and  its  complete 
integral,  which  contains  a  second  constant,  is  also  the  '*  com- 
plete integral "  of  the  given  equation. 

A  differential  equation  of  the  second  order  is  "  exact  "  when, 
all  its  terms  being  transposed  to  the  first  member,  that  member 
is  the  derivative  with  respect  to  x  of  an  expression  of  the  first 
order,  that  is,  a  function  of  x^  y  and  p.  It  is  obvious  that  the 
terms  containing  the  second  derivative,  in  such  an  exact  differ- 
ential, arise  solely  from  the  differentiation  of  the  terms  con- 
taining/ in  the  function  of  x,y  and/.  For  example,  let  it  be 
required  to  ascertain  whether 

is  an  exact  equation.     The  terms  in  question  are  (i  —  x'^)-r-i 

ax 

which  can  arise  only  from  the  differentiation  of  (i  —  x'^^p. 
Now  subtract  from  the  given  expression  the  complete  deriva- 
tive of  (i  —  x^)p,  which  is 

the  remainder  i?  x-^  -\-  y,  which  is  an  exact  derivative,  namely, 
ax 

that  of  xy.  Hence  the  given  expression  is  an  exact  differ- 
ential, and 

(\-x')^J-^xy  =  c,  (6) 

is  the  first  integral  of  the  given  equation.  Solving  this  linear 
equation  for  j,  we  find  the  complete  integral 

7  =  ^,^  +  ^,V(i  -^»).  (7) 

Prob.  55.  Solve  (i  -  x^)"^^  _  ^^  =  2. 

(Ans.  y  =  (sin~^  ^)'  +  c^  sin"^  x  +  ^,.) 


Art.  14.]  THE  TWO  first  integrals.  333 

Prob.  56.  Solve  0  =  ^.  (ahs.  J  =  ^  +  v'-) 

Prob.  57.  Solve  -~-^  =  «'^  —  /^y 

(Ans.  a^x  —  b'^y  =  A  sin  ^^  +  ^  cos  bx.) 
Prob.  58.  Solve  ^g  +  (^)'  =  i.     (Ans.  /  =  ^'^  +  c,x  +  .,.) 


Art.  14.    The  Two  First  Integrals. 

We  have  seen  in  the  preceding  article  that  the  complete 
integral  of  an  equation  of  the  second  order  is  a  relation  be- 
tween X,  y  and  two  constants  c,  and  c^ .  Conversely,  any  rela- 
tion between  x,  y  and  two  arbitrary  constants  may  be  regarded 
as  a  primitive,  from  which  a  differential  equation  free  from  both 
arbitrary  constants  can  be  obtained.  The  process  consists  in 
first  obtaining,  as  in  Art.  3,  a  differential  equation  of  the  first 
order  independent  of  one  of  the  constants,  say  c^,  that  is,  a  rela- 
tion between  x,  y,p  and  c^ ,  and  then  in  like  manner  eliminating 
f,  from  the  derivative  of  this  equation.  The  result  is  the  equa- 
tion of  the  second  order  or  relation  between  x,  y,  p  and  q  {q 
denoting  the  second  derivative),  of  which  the  original  equation 
is  the  complete  primitive,  the  equation  of  the  first  order  being 
the  first  integral  in  which  r,  is  the  constant  of  integration.  It 
is  obvious  that  we  can,  in  like  manner,  obtain  from  the  primi- 
tive a  relation  between  x,  y,  p  and  c^ ,  which  will  also  be  a  first 
integral  of  the  differential  equation.  Thus,  to  a  given  form  of 
the  primitive  or  complete  integral  there  corresponds  two  first 
integrals. 

Geometrically  the  complete  integral  represents  a  doubly 
infinite  system  of  curves,  obtained  by  varying  the  values  of  c^ 
and  of  c^  independently.  If  we  regard  c^  as  fixed  and  c^  as 
arbitrary,  we  select  from  that  system  a  certain'  singly  infinite 
system;  the  first  integral  containing  r,  is  the  differential  equa- 
tion of  this  system,  which,  as  explained  in  Art.  2,  is  a  relation 
between  the  coordinates  of  a  moving  point  and  the  direction 
of  its  motion  common  to  all  the  curves  of  the  system.     But 


:334 


DIFFERENTIAL    EQUATIONS. 


[Chap.  VII. 


the  equation  of  the  second  order  expresses  a  property  involv- 
ing curvature  as  well  as  direction  of  path,  and  this  property 
•being  independent  of  c^  is  common  to  all  the  systems  corre- 
sponding to  different  values  of  c^,  that  is,  to  the  entire  doubly 
infinite  system.  A  moving  point,  satisfying  this  equation, 
may  have  any  position  and  move  in  any  direction,  provided  its 
path  has  the  proper  curvature  as  determined  by  the  value  of  q 
derived  from  the  equation,  when  the  selected  values  of  x^  y 
2iV\(\  p  have  been  substituted  therein.* 

For  example,  equation  (7)  of  the  preceding  article  repre- 
sents an  ellipse  having  its  center  at  the  origin  and  touching 
the  lines  ;ir  =  di  i,  as  in  the  diagram  ;  c^  is  the  ordinate  of  the 
point  of  contact  with  ;r  =  i,  and  c^  that  of  the  point  in  which 
the  ellipse  cuts  the  axis  oi  y.  If  we  regard  c^  as  fixed  and  c^ 
as  arbitrary,  the  equation  represents  the  system  of  ellipses 
touching  the  two  lines  at  fixed  points,  and  equation  (6)  is  the 
differential  equation  of  this  system.  In 
like  manner,  if  c^  is  fixed  and  c^  arbitrary, 
equation  (7)  represents  a  system  of  ellipses 
cutting  the  axis  of  y  in  fixed  points 
and  touching  the.  lines  x  =  ±  i.  The 
corresponding  differential  equation  will  be 
found  to  be 

Pinally,  the  equation  of  the  second  order,  independent  of  ^, 
and  c^  [(5)  of  the  preceding  article]  is  the  equation  of  the 
doubly  infinite  system  of  conies  f  with  center  at  the  origin, 
and  touching  the  fixed  lines  x  ^=  ±  i. 

*  If  the  equation  is  of  the  second  or  higher  degree  in  ^,  the  condition  for 
equal  roots  is  a  relation  between  x,  y  and/,  which  may  be  found  to  satisfy  the 
given  equation.  li  it  does,  it  represents  a  system  of  singular  solutions;  each 
of  the  curves  of  this  system,  at  each  of  its  points,  not  only  touches  but  osculates 
with  a  particular  integral  curve.  It  is  to  be  remembered  that  a  singular  solu- 
tion of  a  first  integral  is  not  generally  a  solution  of  the  given  diflfereniial  equa- 
tion; for  it  represents  a  curve  which  simply  touches  but  does  not  osculate  a  set 
of  curves  belonging  to  the  doubly  infinite  system. 

f  Including  hyperbolas  corresponding  to  imaginary  values  of  c-i. 


Art.  14.]  THE  TWO  first  integrals.  335 

But,  starting  from  the  differential  equation  of  second  order, 
we  may  find  other  first  integrals  than  those  above  which  corre- 
spond to  c^  and  c^.  For  instance,  if  equation  (5)  be  multiplied 
by/,  it  becomes 

dp 

(I  -  ^')/  f^  -  ^f  -^yp  =  o, 

which  is  also  an  exact  equation,  giving  the  first  integral 

(i-^V  +  /  =  C 

in  which  c^  is  a  new  constant  of  integration. 

Whenever  two  first  integrals  have  thus  been  found  inde- 
pendently, the  elimination  of  /  between  them  gives  the  com- 
plete integral  without  further  integration.*  Thus  the  result 
of  eliminating  /  between  this  last  equation  and  the  first  inte- 
gral containing  c^  [equation  (6),  Art.  13]  is 

y  —2c^xj/  +  ^,V  =  c,'  —  ^,', 

which  is  therefore  another  form  of  the  complete  integral.  It 
IS  obvious  from  the  first  integral  above  that  c^  is  the  maximum 
value  of  /,  so  that  it  is  the  differential  equation  of  the  system 
of  elHpse  inscribed  in  the  rectangle  drawn  in  the  diagram.  A 
•comparison  of  the  two  forms  of  the  complete  integral  shows 
that  the  relation  between  the  constants  is  ^3''  =  c/  -(-  <:/. 

If  a  first  integral  be  solved  for  the  constant,  that  is,  put  in 
the  form  (p(x,  y,  p)  =  c,  the  constant  will  disappear  on  differ- 
entiation, and  the  result  will  be  the  given  equation  of  second 
order  multiplied,  in  general,  by  an  integrating  factor.  We  can 
thus  find  any  number  of  integrating  factors  of  an  equation 
already  solved,  and  these  may  suggest  the  integrating  factors 
of  more  general  equations,  as  illustrated  in  Prob.  59  below. 

*  The  principle  of  this  method  has  already  been  applied  in  Art,  10  to  the 
solution  of  certain  equations  of  the  first  order;  the  process  consisted  of  forming 
the  equation  of  the  second  order  of  which  the  given  equation  is  a  first  integral 
-(but  with  a  particular  value  of  the  constant),  then  finding  another  first  integral 
and  deriving  the  complete  integral  by  elimination  of  /. 


336  DIFFERENTIAL   EQUATIONS.  [Chap.  VII- 

73 

Prob.  59.  Solve  the  equation  -7-5  -{-  ay  =  o  in  the  form 

y  =  A  cos  ax  -\-  B  sin  ax; 

and  show  that  the  corresponding  integrating  factors  are  also  inte- 
grating factors  of  the  equation 

where  X  is  any  function  of  x;  and  thence  derive  the  integral  of  this 
equation. 

(Ans.  ay  =  sin  ax  I  cos  ax  .  Xdx  —  cos  ax  I  sin  ax  .  Xdx). 

Prob.  60.  Find  the  rectangular  and  also  the  polar  differential 
equation  of  all  circles  passing  through  the  origin. 


Art.  15.    Linear  Equations. 

A  linear  differential  equation  of  any  order  is  an  equation  of 
the  first  degree  with  respect  to  the  dependent  variable  y  and 
each  of  its  derivatives,  that  is,  an  equation  of  the  form 

where  the  coefBcients  /*„»  •  •  •  -^«  ^^^  the  second  member  A' are 
functions  of  the  independent  variable  only. 

The  solution  of  a  linear  equation  is  always  supposed  to  be 
in  the  form  y  ^f{x) ;  and  if  y^  is  a  function  which  satisfies  the 
equation,  it  is  customary  to  speak  of  the  function  j/,,  rather  than 
of  the  equation  j  =  ^i,  as  an  "integral"  of  the  linear  equa- 
tion. The  general  solution  of  the  linear  equation  of  the  first 
order  has  been  given  in  Art.  6.  For  orders  higher  than  the 
first  the  general  expression  for  the  integrals  cannot  be  effected 
by  means  of  the  ordinary  functional  symbols  and  the  integral 
sign,  as  was  done  for  the  first  order  in  Art.  6. 

The  solution  of  equation  (i)  depends  upon  that  of 


Art.  15.]  linear  equations.  337 

The  complete  integral  of  this  equation  will  contain  71  arbi- 
trary constants,  and  the  mode  in  which  these  enter  the  expres- 
sion for  y  is  readily  inferred  from  the  form  of  the  equation. 
For  let  y^  be  an  integral,  and  c^  an  arbitrary  constant ;  the  re- 
sult of  putting  y  =  cj^  in  equation  (2)  is  c^  times  the  result  of 
putting  y  ^ y^\  that  is,  it  is  zero ;  therefore  c^y^  is  an  integral. 
So  too,  if  y^  is  an  integral,  c^^  is  an  integral ;  and  obviously 
also  c^y^  -f-  c^y^  is  an  integral.  Thus,  if  n  distinct  integrals/,, 
y^i-  '  ' yn  can  be  found, 

y  =  c,y,  +  c^y^  +  . . .  +  Cnyn  (3) 

will  satisfy  the  equation,  and,  containing,  as  it  does,  the  proper 
number  of  constants,  will  be  the  complete  integral. 

Consider  now  equation  (i);  let  Fbe  a  particular  integral  of 
it,  and  denote  by  u  the  second  member  of  equation  (3),  which 
is  the  complete  integral  when  X  =  o.     If 

y=Y+u  (4) 

be  substituted  in  equation  (i),  the  result  will  be  the  sum  of  the 
results  of  putting  y  =  Fand  of  putting  y  ^  u  ;  the  first  of 
these  results  will  be  ^,  because  Fis  an  integral  of  equation  (i), 
and  the  second  will  be  zero  because  u  is  an  integral  of  equa- 
tion (2).  Hence  equation  (4)  expresses  an  integral  of  (2);  and 
since  it  contains  the  n  arbitrary  constants  of  equation  (3),  it 
is  the  complete  integral  of  equation  (i).  With  reference  to 
this  equation  F  is  called  "  the  particular  integral,"  and  u  is 
called  ''the  complementary  function."  The  particular  integral 
contains  no  arbitrary  constant,  and  any  two  particular  integrals 
may  differ  by  any  multiple  of  a  term  belonging  to  the  comple- 
mentary function. 

If  one  term  of  the  complementary  function  of  a  linear 
equation  of  the  second  order  be  known,  the  complete  solution 
can  be  found.  For  let  j/,  be  the  known  term  ;  then,  if  y  =  y^v 
be  substituted  in  the  first  member,  the  coefficient  of  7j  in  the 
result  will  be  the  same  as  if  v  were  a  constant :  it  will  there- 
fore be  zero,  and  7j  being  absent,  the  result  will  be  a  linear  equa- 
tion of  the  first  order  for  v\  the  first  derivative  o-f  v.     Under 


338  DIFFERENTIAL    EQUATIONS.  [ChaP.  VII. 

the  same  circumstances  the  order  of  any  linear  equation  can 
in  like  manner  be  reduced  by  unity. 

A  very  simple  relation  exists  between  the  coefficients  of  an 
exact  linear  equation.  Taking,  for  example,  the  equation  of 
the  second  order,  and  indicating  derivatives  by  accents,  if 

is  exact,  the  first  term  of  the  integral  will  be  Pj'  Subtracting 
the  derivative  of  this  from  the  first  member,  the  remainder  is 
(P^  —  P^)y'  +  P^y.  The  second  term  of  the  integral  must 
therefore  be  {P^  —  Po)y ;  subtracting  the  derivative  of  this  ex- 
pression, the  remainder,  (P^  —  P/  -\-  P/')y,  must  vanish.  Hence 
P^  —  P/  -\-  P^'  =  o  is  the  criterion  for  the  exactness  of  the 
given  equation.  A  similar  result  obviously  extends  to  equa- 
tions of  higher  orders. 

Prob.  6i.  Solve  x— (3  +  x)-J-  -f  3^  =  o,  noticing  that  e=^  is 

an  integral.  (Ans.  y  =  c/"  +  cj^x^  +  ^x^  +  6^  +  6.) 

72  J 

Prob.  62.  Solve  (x^  —  ^)-pi  +  ^{^x  +  1)-^  -\-  2y  =  o. 

(Ans.   (x  —  i)^^  =  cX^*'  —  6^^  -\-  2X  —  \  —  4x^  log  x)  +  c^x^.) 

Prob.  62,'  Solve-3^  +  cos  0-T^  —  2  sin  6-^  —y  cos  6  =  sin  26. 


^Ans.  y  =  ^-si"^y*^si"«((:,<9  +  c^d -}- 


c^e 


sin0  


sin  6^  —  I 


Art.  16.    Linear  Equations  with  Constant 
Coefficients. 

The  linear  equation  with  constant  coefficients  and  second 
member  zero  may  be  written  in  the  form 

A,D-y  +  A,D-'y  +  , .  .  +  A„y  =  (i) 

in  which  I)  stands  for  the  operator  ^,  D^  for  -^j,  etc.,  so  that 

V  indicates  that  the  operator  is  to  be  applied  n  times.     Then, 
;since  Z^^'"^  =  me'"'',  D'^e'*'''  =  wV'"^,  etc.,  it   is  evident   that    if 


Art.  16.]      linear  equations,  constant  coefficients.  339 

y  =  e""""  be  substituted  in  equation  (l),  the  result  after  rejecting 
the  factor  e*^""  will  be 

^^w"  +  A.m''-'  -{-..,-{- A„  =  o.  (2) 

Hence,  if  m  satisfies  equation  (2),  e'"''  is  an  integral  of  equation 
(i) ;  and  if  m^,  m^,  . .  .  m„  are  n  distinct  roots  of  equation  (2), 
the  complete  integral  of  equation  (i)  will  be 

y  =  c.e^i-  +  c,e^n^^  +  .  . .  +  ^ne"*'"'.  (3) 

For  example,  if  the  given  equation  is 
d^y       dy 

the  equation  to  determine  m  is 

m^  —  m  —  2  ^O, 

of  which  the  roots  are  ;;^,  =  2,  w,  =  —  i  ;  therefore  the  in- 
tegral is 

y  =  c^e'''  +  ^a^"*- 

The  general  equation  (i)  may  be  written  in  the  symbolic 
form  f{D)  .y  =  o,  in  which  /  denotes  a  rational  integral  func- 
tion. Then  equation  (2)  is  /{m)  =  o,  and,  just  as  this  last 
equation  is  equivalent  to 

{m  —  m^)(m  —  w,) . , ,  {m  —  m„)  =  o,  (4) 

so  the  symbolic  equation /{D).y  =  o  may  be  written 

{D  —  m,)(D  —  m^  . .  .  (D  —  m^y  =  o.  (5) 

This  form  of  the  equation-shows  that  it  is  satisfied  by  each  of 
the  quantities  which  satisfy  the  separate  equations 

{D  —  m,)y  =  o,     (D  —  m^y  =  0  . .  .{D  —  m„)y  =  o\       (6) 

that  is  to  say,  by  the  separate  terms  of  the  complete  integral. 

If  two  of  the  roots  of  equation  (2)  are  equal,  say  to  m^,  two 
of  the  equations  (6)  become  identical,  and  to  obtain  the  full 
number  of  integrals  we  must  find  two  terms  corresponding  to 
the  equation 

(D-m,yy  =  o^  (7) 

in  other  words,  the  complete  integral  of  this  equation  of  which 
y^  =  ^'^1*  is  known  to  be  one  integral.     For  this  purpose  we 


340  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

put,  as  explained  in  the  preceding  article,  jj/  =jKi^.  By  differen- 
tiation, Dy  —  De'^^^'v  =  e'^^'^ijit-v  -\-  Dv)  ;  therefore 

(P  —  7^  J^'^i^z;  =  e^'^^'Dv.  (8) 

In  like  manner  we  find 

{D  —  ;;2,)V'«i^z'  =  e'^^'^Vv.  (9) 

Thus  equation  (7)  is  transformed  to  D'v^^o,  of  which  the 
complete  integral  is  2^  =  c^x  -\-c^ ;  hence  that  of  equation  (7)  is 

y  —  e'^^^'ic^x-^c^,    -  (io> 

These  are  therefore  the  two  terms  corresponding  to  the  squared 
factor  {D  —  m^  in  f{D)y  =  o. 

It  is  evident  that,  in  a  similar  manner,  the  three  terms 
corresponding  to  a  case  of  three  equal  roots  can  be  shown  ta 
be  c'"^''(c^x''  -\-  c^x  -j-  ^^3),  and  so  on. 

The  pair  of  terms  corresponding  to  a  pair  of  imaginary- 
roots,  say  ;«j  :=  a -\-  ijS,  m^=.  a  —  ift,  take  the  imaginary  form 

Separating  the  real  and  imaginary  parts  of  e'^''  and  ^-*^^,  and 
changing  the  constants,  the  expression  becomes 

^'^(^  cos /?jr  +  i5  sin  y5;i;).  (li) 

For  a  multiple  pair  of  imaginary  roots  the  constants  A  and 
B  must  be  replaced  by  polynomials  as  above  shown  in  the  case 
of  real  roots. 

When  the  second  member  of  the  equation  with  constant   ♦ 

coefificients  is  a  function  of  X,  the  particular  integral  can  also 

be  made  to  depend  upon  the  solution  of  hnear  equations  of 

the  first   order.     In   accordance  with   the   symbolic   notation 

introduced  above,  the  solution  of  the  equation 

-     dy 

—  -ay  =  X,     or     (D  -  a)y  =  X  (12) 

is  denoted  by  y  =  {D  —  a)-'X,  so  that,  solving  equation  (12), 
we  have 

-^^X=^\fe-"Xdx  (13) 

as  the  value  of  the  inverse  symbol  whose  meaning  is  "that 


Art.  16.]        linear  equations,  constant  coefficients.  341 

function  of  x  which  is  converted  to  X  by  the  direct  operation 
expressed  by  the  symbol  D  —  ^."  Taking  the  most  convenient 
special  value  of  the  indefinite  integral  in  equation  (13),  it  gives 
the  particular  integral  of  equation  (12).  In  like  manner,  the  par- 
ticular integral  oi  f{P)y  ^^  X  is  denoted  by  the  inverse  symbol 

jtjyX.     Now,  with  the  notation  employed  above,  the  symbolic 

fraction  may  be  decomposed  into  partial  fractions  with  constant 
numerators  thus : 

\  N  N  N 

Trj^^X  =  n      '      ^+  n      '      ^  +  .  •  •  +  TT^^^^""  (H) 
/{IJ)  D  —  ;;/,       '   D  —  m^       '  ^    D  —  m^        ^  ^' 

in  which  each  term  is  to  be  evaluated  by  equation  (13),  and 
may  be  regarded  (by  virtue  of  the  constant  involved  in  the 
indefinite  integral)  as  containing  one  term  of  the  complement- 
ary  function.     For   example,   the    complete    solution    of   the 

equation 

^ jj/  _dy  _       _  y 

dx^       dx 
is  thus  found  to  be 

y  =  \^Je-^^Xdx  -  \e-^fe^Xdx. 

When  X  is  a  power  of  x  the  particular  integral  may  be 
found  as  follows,  more  expeditiously  than  by  the  evaluation  of 
the  integrals  in  the  general  solution.  For  example,  \{  X  —  x"" 
the  particular  integral  in  this  example  may  be  evaluated  by 
development  of  the  inverse  symbol,  thus  : 

y  -  D'-  D-2'~      21  ^\{D-  D'f 

=  - i[i  -  i(Z)  -  D^)  +  \{D  -  DJ  -  .  .  ,-\x' 
=  -lli-^D  +  iD^-  . .  .>'  =  _i^'  + J^'_|. 

*  The  validity  of  this  equation  depends  upon  the  fact  that  the  operations 
expressed  in  the  second  member  of 

f{D)  =  (D-  m,\D  -m^)-\-...-^r{D-mn) 
are  commutative,  hence  the  process  of  verification  is  the  same  as  if  the  equation 
were  an  algebraic  identity.  This  general  solution  was  published  by  Boole  in 
the  Cambridge  Math.  Journal,  First  Series,  vol.  ii,  p.  114.  It  had,  however, 
been  previously  published  by  Lobatto,  Th6orie  des  Characteristiques,  Amster- 
dam, 1837. 


342  DIFFERENTIAL    EQUATIONS.  [ChAP.  VII. 

The  form  of  the  operand  shows  that,  in  this  case,  it  is  only 
necessary  to  carry  the  development  as  far  as  the  term  contain- 
ing  D\ 

For  other  symbolic  methods  applicable  to  special  forms  of 
Xwe  must  refer  to  the  standard  treatises  on  this  subject. 

d^y        dy 
Prob.64.  Solve  4^3 -37^+^  =  0. 

(Ans.  y  =  e^(Ax  -{- B) -\-  a-^\) 

Prob.  65.   Show  that     — ^^"^  =  -J-^^'"'' 
^  AD)  f{a) 

and  that     777^  sin  {ax  +  /?)  =  — sin  {ax  -\-  /?). 

Prob.  66.  Solve  {IT  +  i)j'  =  ^  +  sin  2x -\-  sin  x,  (Compare 
Prob.  59,  Art.  14.) 

(Ans.  y=^  A  sin  ^  +  -^  cos  x  +  i^  —  -k  sin  2x  —  ^x  cos  x.) 


Art.  17.    Homogeneous  Linear  Equations. 
The  linear  difTerential  equation 

in  which  A^,  A^,  etc.,  are  constants,  is  called  the  "homogene- 
ous linear  equation."  It  bears  the  same  relation  to  x""  that 
the  equation  with  constant  coefficients  does  to  e'"''.  Thus,  if 
yz=x'*'  be  substituted  in  this  equation,  the  factor  x"*"  will  divide 
out  from  the  result,  giving  an  equation  for  determining  #/, 
and  the  n  roots  of  this  equation  will  in  general  determine  the 
n  terms  of  the  complete  integral.  For  example,  if  in  the 
equation 

^  Vt  +  2^T^  —  27  =  O 

dx  dx 

we  put  y  =  x^^  the  result  is  m{m  —  i)  +  2m  —  2  =  0,  or 
{m  —  \){in  +  2)  =  o. 

The   roots   of   this   equation    are   m^  =  i    and   m^  =  —  2. 
Hence  y  =  c^x  -\-  c^x~^ 

is  the  complete  integral. 

Equation  (i)  might  in  fact  have  been  reduced  to  the  form 
with  constant  coefficients  by  changing  the  independent  vari= 


Art.  17.]  homogeneous  linear  equations.  343 

able  to  <9,  where  ;i:  =;  ^^,  or  6  ^  log  x.  We  may  therefore  at 
once  infer  from  the  results  established  in  the  preceding  article 
that  the  terms  corresponding  to  a  pair  of  equal  roots  are  of  the 
form 

{C,  +  C,  log  X)X^'\  (2) 

and  also  that  the  terms  corresponding  to  a  pair  of  imaginary 
roots,  a  di  ifi,  are 

x'^lA  cos  (/5  log  x)-\-B  sin  {ft  log  x)].  (3) 

The  analogy  between  the  two  classes  of  linear  equations 
considered  in  this  and  the  preceding  article  is  more  clearly 
seen  when  a  single  symbol  0==  xD  is  used  for  the  operation  of 
taking  the  derivative  and  then  multiplying  by  x,  so  that 
■^x'*"  =  inx"\  It  is  to  be  noticed  that  the  operation  x'^D'^  is  not 
the  same  as  t9'  or  xDxD,  because  the  operations  of  taking  the 
derivative  and  multiplying  by  a  variable  are  not  ''commu- 
tative/' that  is,  their  order  is  not  indifferent.  We  have,  on  the 
contrary,   x^D^  =  -Bip  —  i)  ;    then   the    equation   given    above, 

which  is 

{x''D'^2xD  -2)y  =  o, 
becomes 

[^(^  —  i)  +  2^  —  2]y  =  o,     or     (^  -  i)(^  +  2)y  =  o, 

the  function  of  -&  produced  being  the  same  as  the  function  of 
m  which  is  equated  to  o  in  finding  the  values  of  m. 

A  linear  equation  of  which  the  first  member  is  homoge- 
neous and  the  second  member  a  function  of  x  may  be  reduced 
to  the  form 

A&).y^X.  (4) 

The  particular  integral  may,  as  in  the  preceding  article  (see 
eq.  (14)),  be  separated  into  parts  each  of  which  depends  upon 
the  solution  of  a  linear  equation  of  the  first  order.  Thus, 
solving  the  equation 

;,±-ay  =  X,     or     {^  -  a)y  =  X,  (5) 

we  find  -K-^  X  =  x^  fx-^-'Xdx,  (6) 

The   more   expeditious  method   which   may  be   employed 


344  DIFFERENTIAL    EQUATIONS.  [ChaP.  VII. 

when  Xis  2i  power  of  x  is  illustrated  in  the  following  example : 

dx^         dx 
geneous  when  multiplied  by  x^  and  the  reduced  equation  is 

The  roots  of /(t&)  =o  are  3  and  the  double  root  zero,  hence 
the  complementary  function  is  c^x^ -\- c^-\- c^\o^  x.     Since  in 
general /(i^);!;*"  =:/(r);ir'',  we  infer  that  in  operating  upon  x^  we 
may  put  i&  =  3.     This  gives  for  the  particular  integral 
I       i_^,       I       I        , 


but  fails  with  respect  to  the  factor  ^  —  3.*     We  therefore 
now  fall  back  upon  equation  (6),  which  gives 

x^  —  x^  I  x'^dx  =  x^  log  X. 

^  —  3  *^ 

The  complete  integral  therefore  is 

7  =  c,x'  +  c,  +  c,  log  X  +  -i^'  log  X, 

d'^y           dy 
Prob.  67.  Solve  '^x'-p,  +  3-^ 7 3^  =  •^'• 

(Ans.  y  =i  c^x-\-  c^x-%  +  ^j;'.) 
Prob.  68.  Solve  {x'D'  +  3^Z>'  +  D)y  =  ^. 

(Ans.  y  =  c,-]-c^\ogx-[-  c^(\og  x)'  +  ^(log  x)'\) 

Art.  18.    Solutions  in  Infinite  Series. 

We  proceed  in  this  article  to  illustrate  the  method  by 
which  the  integrals  of  a  linear  equation  whose  coefficients  are 
algebraic  functions  of  x  may  be  developed  in  series  whose 
terms  are  powers  of  x.  For  this  purpose  let  us  take  the 
equation 

*  The  failure  occurs  because  x'^  is  a  term  of  the  complementary  function 
having  an  indeterminate  coefficient;  accordingly  the  new  term  is  of  the  same 
form  as  the  second  term  necessary  when  3  is  a  double  root,  but  of  course  with 
a  determinate  coefficient. 


Art.  18]  SOLUTIONS    IN    INFINITE    SERIES.  345 

which  is  known  as  *'  Bessel's  Equation,"  and  serves  to  define 
the  ''Besselian  Functions."  -*  - — - 

If  in  the  first  member  of  this  equation  we  substitute  for  j 
the  single  term  Ax""  the  result  is 

^(w''-;2>'«^-^;lr'«+^  (2) 

the  first  term  coming  from  the  homogeneous  terms  of  the 
equation  and  the  second  from  the  term  x'^y  which  is  of  higher 
■degree.  If  this  last  term  did  not  exist  the  equation  would  be 
satisfied  by  the  assumed  value  oi  y,  if  m  were  determined  so  as 
to  make  the  first  term  vanish,  that  is,  in  this  case,  by  Ax""  or 
Bx~^.  Now  these  are  the  first  terms  of  two  series  each  of 
which  satisfies  the  equation.  For,  if  we  add  to  the  value  of  y 
a  term  containing  ;r'"+%  thus  j/  =  A^x""  -|-  ^,;ir*"+^  the  new  term 
will  give  rise,  in  the  result  of  substitution,  to  terms  containing 
j^m+2  ^j^^  ^m+4  respectively,  and  it  will  be  possible  so  to  take 
A^  that  the  entire  coefficient  of  x"''^''  shall  vanish.  In  like 
manner  the  proper  determination  of  a  third  term  makes  the 
coefficient  of  x"'+'^  in  the  result  of  substitution  vanish,  and  so 
on.     We  therefore  at  once  assume 

y=2 A^'^^'*-  =  AX'  +  Ai ^'""^'  +  A^x'^+*'+  .  .  .  ,       (3) 
0 

in  which  r  has  all  integral  values  from  o  to  00 .     Substituting 

in  equation  (i) 

2[{{m  +  2rf-  n'\A^'^+''+  ^^'«+^(^+'^]  =  o.  (4) 

The  coefficient  of  each  power  of  x  in  this  equation  must  sep- 
arately vanish  ;  hence,  taking  the  coefficient  of  ;ir'"+^'',  we  have 

[{m  +  2ry  -  n']A^  +  A^.,  =  o.  (5) 

When  r  =  6,  this  reduces  to  ;;/"  —  n^  =  o,  which  determines 
the  values  of  m,  and  for  other  values  of  r  it  gives 

,^.-x,  (6) 


''  ~        (m  -j-  2r-\-  n){m  -\-  2r  —  n) 
the  relation  between  any  two  successive  coefficients. 

For  the  first  value  of  in,  namely  Uy  this  relation  becomes 

A  '  A     ■ 

'~       2\n-\-r)r     •■-'" 


346     ,  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

whence,  determining  the  successive  coefficients  in  equation  (3), 
the  first  integral  of  the  equation  is 

Ao',  =  A,^\i  -  ;^  7,  +  („+,x^^,)  i^  -  •  •  •]•  (7) 

In  Hke  manner,  the  other  integral  is  found  to  be 
B,y,  =  B,.-'  [i  +  -i-  g  +  ^^^  _  ^^^^  _  ,^  ,-4;  +  ■  •  ■]■  (8) 

and  the  complete  integral  is  j^  =  A^^  -\-  Bj^.^ 

This  example  illustrates  a  special  case  which  may  arise  in 
this  form  of  solution.  If  «  is  a  positive  integer,  the  second 
series  will  contain  infinite  coefficients.  For  example,  if  it  =  2, 
the  third  coefficient,  or  B^,  is  infinite,  unless  we  take  B^  =z  o,  in 
which  case  B^  is  indeterminate  and  we  have  a  repetition  of  the 
solution  jKi-  This  will  always  occur  when  the  same  powers  of 
X  occur  in  the  two  series,  including,  of  course,  the  case  in  which 
m  has  equal  roots.  For  the  mode  of  obtaining  a  new  integral 
in  such  cases  the  complete  treatises  must  be  referred  to.f 

It  will  be  noticed  that  the  simplicity  of  the  relation  between 
consecutive  coefficients  in  this  example  is  due  to  the  fact  that 
equation  (i)  contained  but  two  groups  of  terms  producing 
different  powers  of  x,  when  Ax""  is  substituted  for  j^  as  in  ex- 
pression (2).  The  group  containing  the  second  derivative 
necessarily  gives  rise  to  a  coefficient  of  the  second  degree  in 
7n,  and  from  it  we  obtained  two  values  of  m.  Moreover,  be- 
cause the  other  group  was  of  a  degree  higher  by  two  units,  the 
assumed  series  was  an  ascending  one,  proceeding  by  powers 
of  x\ 

*  The  Besselian  function  of  the  wth  order  usually  denoted  by /«  is  the  value 
of  jj  above,  divided  by  2»n\  if  n  is  a  positive  integer,  or  generally  by  2**F(«+i). 
For  a  complete  discussion  of  these  functions  see  Lommel's  Studien  iiber  die 
Bessel'schen  Functionen,  Leipzig,  1868;  Todhunter's  Treatise  on  Laplace's, 
Lamp's  and  Bessel's  Functions,  London,  1875,  etc. 

f  A  solution  of  the  kind  referred  to  contains  as  one  term  the  product  of  the 
regular  solution  and  log  x,  and  is  sometimes  called  a  "  logarithmic  solution." 
See  also  American  Journal  of  Mathematics,  Vol.  XI.  p.  37.  In  the  case  of 
Bessel's  equation,  the  logarithmic  solution  is  the  "Besselian  Function  of  the 
second  kind." 


Art.  18.]  SOLUTIONS  in  infinite  series.  347 

In  the  following  example, 

d'^y         dy  y 

there  are  also  two  such  groups  of  terms,  and  their  difference 
of  degree  shows  that  the  series  must  ascend  by  simple  powers. 
We  assume  therefore  at  once 

7  =  l^,;t^'«+^  (lO) 

0 

The  result  of  substitution  is 

S[{(w+r)(w+r-i)-2M^'«+'^^  +  ^(w+r)^^;j;'«+*-^]=:  o.  (ii^ 

Equating  to  zero  the  coefficient  of  ;i;*"+''"^, 

(w  +  r  +  \){m  +  r  —  2)Ar  +  a{pt  +  ^  ~  l)-^*""'  =  O,  (12) 
which,  when  r  =  o,  gives 

(;;^+i)(;«-2)^„=o,  (13) 

and  when  r  >  o, 

_         ni-\-r  —  \ 

The  roots  of  equation  (13)  are  m=  2  and  m  =.  —  i\  taking 
m  =  2,  the  relation  (14)  becomes 

A   -       a"^^^  A 

whence  the  first  integral  is 

Taking  the  second  value  m  =  —  i,  equation  (14)  gives 

K^  —  3) 

a 
whence  B^=-  —  -B^ ,  and  ^^  =  o  *;  therefore  the  second  inte- 
gral is  the  finite  expression 

B^y,^B,.-\,-\a.'\-=Bl}^-\\.  (l6) 

*  Bz  would  take  the  indeterminate  form,  and  if  we  suppose  it  to  have  a  finite 
value,  the  rest  of  the  series  is  equivalent  to  B%y\,  reproducing  the  first  integral. 


548  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

When  the  coefficient  of  the  term  of  highest  degree  in  the 
result  of  substitution,  such  as  equation  (ii),  contains  7n,  it  is 
possible  to  obtain  a  solution  in  descending  powers  of  x.  In 
this  case,  m  occurring  only  in  the  first  degree,  but  one  such 
solution  can  be  found  ;  it  would  be  identical  with  the  finite 
integral  (i6).  In  the  general  case  there  will  be  two  such  solu- 
tions, and  they  will  be  convergent  for  values  of  x  greater  than 
unity,  while  the  ascending  series  will  converge  for  values  less 
than  unity.* 

When  the  second  member  of  the  equation  is  a  power  of  x, 
the  particular  integral  can  be  determined  in  the  form  of  a  series 
in  a  similar  manner.  For  example,  suppose  the  second  mem- 
ber of  equation  (9)  to  have  been  x^.  Then,  making  the  sub- 
stitution as  before,  we  have  the  same  relation  between  consecu- 
tive coefficients;  but  when  r  =  o,  instead  of  equation  (13)  we 

have 

{m  +  \){ni  —  2)A,x'''-'  =  xi 

to  determine  the  initial  term  of  the  series.  This  gives  m  =  2^ 
and  Aq  =  ^;  hence,  putting  7n  =  ^  in  equation  (14),  we  find  for 
the  particular  integral  f 


^x^ 

7 


I  _  lllax  4-  -^l^-^^V  -  .  .  .1 
1 1 .  3  .  5  J 


9-3  9-  II. 3. 5 

A  linear  equation  remains  linear  for  two  important  classes 
of  transformations;  first,  when  the  independent  variable  is 
changed  to  any  function  of  x,  and  second,  when  for  j/  we  put 
v/{x).  As  an  example  of  the  latter,  let  jy  =  e'^'^'v  be  substituted 
in    equation  (9)  above.     After   rejecting   the   factor  ^"*'',  the 

result  is 

(Tv         dv       2v  _ 

dx''         dx       x"" 
Since  this  differs  from  the  given  equation  only  in  the  sign 

*When  there  are  two  groups  of  terms,  the  integrals  are  expressible  in  terms 
of  Gauss's  '*  Hypergeometric  Series." 

f  If  the  second  member  is  a  term  of  the  complementary  function  (for  ex- 
ample, \n  this  case,  if  it  is  any  integral  power  ol  x),  the  particular  integral  will 
take  the  logarithmic  form  referred  to  in  the  foot-note  on  p.  346. 


Art.  19.]  SYSTEMS    OF    DIFFERENTIAL    EQUATIONS.  349' 

of  a,  we  infer  from  equation  (i6)  that  it  has  the  finite  integral. 

V  =.  --^  —.     Hence  the  complete  integral  of  equation  (9)  can. 

be  written  in  the  form 

xy  =.  c^(2  —  ax)  4-  c^e~'^''{2  -\-  ax), 

Prob.  69.  Integrate  in  series  the  equation  -j-^  -\-  xy  ^^  o. 

(Ans.  ^=:^(,-i.'+'-V-  .  .  .)+^(.- J.'+^J.'-  .  .  .).) 

d'^y         .  dv 
Prob.  70.   Integrate  in  series  -^'^i— 3  +  ^^-j  +  (-^  ~  2)^  =  o. 

Prob.    71.    Derive    for   the    equation    of    Prob.   70   the  integral 
y^  =  e  ''{x'^  +  I  +  i^),  and  find  its  relation  to  those  found  above. 


Art.  19.    Systems  of  Differential  Equations. 

It  is  shown  in  Art.  12  that  a  determinate  system  of  n  differ- 
ential equations  of  the  first  order  connecting  n  -\-  i  variables 
has  for  its  complete  solution  as  many  integral  equations  con- 
necting the  variables  and  also  involving  n  constants  of  inte- 
gration. The  result  of  eliminating  n  —  i  variables  would  be  a 
single  relation  between  the  remaining  two  variables  containing 
in  general  the  n  constants.  But  the  elimination  may  also  be 
effected  in  the  differential  system,  the  result  being  in  general 
an  equation  of  the  nth  order  of  which  the  equation  just  men- 
tioned is  the  complete  integral.  For  example,  if  there  were 
two  equations  of  the  first  order  connecting  the  variables  x  and 
y  with  the  independent  variable  t,  by  differentiating  each  we 
should  have  four  equations  from  which  to  eliminate  one  vari- 
able,  sayjj/,  and  its  two  derivatives*  with  respect  to  t,  leaving 
a  single  equation  of  the  second  order  between  x  and  t. 

It  is  easy  to  see  that  the  same  conclusions  hold  if  some  of 
the  given  equations  are  of  higher  order,  except  that  the  order 
of  the  result  will  be  correspondingly  higher,  its  index  being  in 

*  In  general,  there  would  be  n^  equations  from  which  to  eliminate  n  —  i 
variables  and  n  derivatives  of  each,  that  is,  {n  —  i){n  -}-  i)  =  «*  —  i  quantities 
leaving  a  single  equation  of  the  wth  order. 


350 


DIFFERENTIAL    EQUATIONS. 


[Chap.  VII. 


general  the  sum  of  the  indices  of  the  orders  of  the  given  equa- 
tions. The  method  is  particularly  applicable  to  linear  equa- 
tions with  constant  coefficients,  since  we  have  a  general  method 
of  solution  for  the  final  result.  Using  the  symbolic  notation, 
the  differentiations  are  performed  simply  by  multiplying  by 
the  symbol  D^  and  therefore  the  whole  elimination  is  of  exactly 
the  same  form  as  if  the  equations  were  algebraic.  For  ex- 
ample, the  system 


(Ty      dx 


dx   ,     dy 


2ny  +  {4l)-s)xz=o; 


y  = 


2/ 

o 


when  written  symbolically,  is 

[t-D"  -  4)y  -  Dx  =  2/, 

whence,  eliminating  Xy 

2n'-4       -D 

2D        aD-  , 

which  reduces  to 

(Z)-i)X2ZP  +  3)>'  =  2-f^. 
Integrating, 

y  =  {A-\-Bt)e'-\-Ce-^'-it, 

the  particular  integral  being  found  by  symbolic  development, 
as  explained  at  the  end  of  Art.  i6. 

The  value  of  x  found  in  like  manner  is 

x  =  {A'  +  B'ty  +  Ce-^'  -  f 

The  complementary  function,  depending  solely  upon  the  deter- 
minant of  the  first  members,*  is  necessarily  of  the  same  form 
as  that  for  y,  but  involves  a  new  set  of  constants.  The  re- 
lations betv/een  the  constants  is  found  by  substituting  the 
values  of  x  and  jj/  in  one  of  the  given  equations,  and  equating 
to  zero  in  the  resulting  identity  the  coefficients  of  the  several 
terms  of  the  complementary  function.  In  the  present  ex- 
ample we  should  thus  find  the  value  of  x,  in  terms  of  Ay  B^ 
and  C,  to  be 

x  =  {6B--2A-  2Bty  —  \Ce-^^'  -  f 

*  The  index  of  the  degree  in  D  of  this  determinant  is  that  of  the  order  of 
the  final  equation  ;  it  is  not  necessarily  the  sum  of  the  indices  of  the  orders  of 
the  given  equations,  but  cannot  exceed  this  sum. 


Art.  19.]  SYSTEMS    OF    DIFFERENTIAL    EQUATIONS.  351 

In  general,  the  solution  of  a  system  of  differential  equations 
depends  upon  our  ability  to  combine  them  in  such  a  way  as 
to  form  exact  equations.  For  example,  from  the  dynamical 
system 

-d?-^'         df-^'         df-^'  ^'^ 

where  X,  V,  Z  are  functions  of  x^  y,  and  2,  but  not  of  /, 
we  form  the  equation 

dx  jdx  ,   dy  jdz    ,  dz  .dz       ^,     1    tz^     i    ^j 
--  d—r  +  -f  ^  ~  +  -rd-T  —  ^dx  +  Ydy  +  Zdz, 
dt    dt^ dt    dt    ^   dt    dt  -^    ' 

The  first  member  is  an  exact  differential,  and  we  know  that  for 
a  conservative  field  of  force  the  second  member  is  also  exact, 
that  is,  it  is  the  differential  of  a  function  U  of  x,  y,  and  z. 
The  integral 


S)+(S) +©>-+-       <») 


is  that  first  integral  of  the  system  (i)  which  is  known  as  the 
equation  of  energy  for  the  unit  mass. 

Just  as  in  Art.  13  an  equation  of  the  second  order  was  re- 
garded as  equivalent  to  two  equations  of  the  first  order,  so  the 
system  (i)in  connection  with  the  equation  defining  the  resolved 
velocities  forms  a  system  of  six  equations  of  the  first  order,  of 
which  system  equation  (2)  is  an  "  integral  "  in  the  sense  ex- 
plained in  Art.  12. 

dx  dy 

Prob.  72.  Solve  the  equations =  —  =  ^/  as  a  system  iin- 

—  my       mx 

ear  in  t.     (Ans.  x  ^=  A  cosmt-\-Bsmmt,y=A  sin mf—B  cosmt.) 

dz  dy 

Prob.  73.  Solve  the  system 1-  7^y  =  e^^  ~   -\-  z  =  o. 

(Ans.  y  =  ^<f«^  +  ^^-"*  +  -^ — ,  z  =  -  nAe"''^  nBe  "  "*  -    f    .) 

n  —  >  n  —  I 

d'^x  d'^y 

Prob.  74.  Find  for  the  system  "^  =  •^0(-^,  >')>  "JpT  —  y^^^^y) 

a  first  integral  independent  of  the  function  0. 

dy         dx  . 

(Ans.x--y~=a) 


362  DIFFERENTIAL   EQUATIONS.  [Chap.  VIL 

Prob.  75.  The  approximate  equations  for  the  horizontal  motion 
of  a  pendulum,  when  the  earth's  rotation  is  taken  into  account,  are 

d^x  dy    ,  s:x  dy    ,         dx    ,  gy 

show  that  both  x  a.ndy  are  of  the  form 

A  cos  nj  +  B  sin  n^t  +  C  cos  n^t  +  -^  sin  «/. 


Art.  20.    First  Order  and  Degree  with  Three 
Variables. 

The  equation  of  the  first  order  and  degree  between  three 
variables  x,  y  and  z  may  be  written 

Pdx  +  Qdy  +  Rdz  =  o,  (i) 

where  P,  Q  and  R  are  functions  of  x,  y  and  z.  When  this 
equation  is  exact,  P,  Q  and  R  are  the  partial  derivatives  of 
some  function  u,  of  x,  y  and  z ;  and  we  derive,  as  in  Art.  4, 

dP^-dQ        'dQ^'dR_       dR^dP^  .. 

dy     a^'      d^      dy'     dx      d2  ^^^ 

for  the  conditions  of  exactness.  In  the  case  of  two  variables, 
when  the  equation  is  not  exact  integrating  factors  always  exist; 
but  in  this  case,  there  is  not  always  a  factor  u  such  that  //P, 
^Q  and  jjR  (put  in  place  of  P,  Q,  and  R)  will  satisfy  all  three 
of  the  conditions  (2).  It  is  easily  shown  that  for  this  purpose 
the  relation 


(3> 


must  exist  between  the  given  values  of  P,  Q,  and  R.     This  is 
therefore  the  "  condition  of  integrability  "  of  equation  (i).* 

When  this  condition  is  fulfilled  equation  (i)  may  be  inte- 
grated by  first  supposing  one  variable,  say  z,  to  be  constant. 
Thus,  integrating  Pdx  +  Qdy  =  o,  and  supposing  the  constant 
of  integration  (7  to  be  a  function  of  z,  we  obtain  the  integral,  so 

*When  there  are  more  than  three  variables  such  a  condition  of  integra- 
bility exists  for  each  group  of  three  variables,  but  these  conditions  are  not  ali 
independent.  Thus  with  four  variables  there  are  but  three  independent  con- 
ditions. 


Art.  30  ]       first  order  and  degree,  three  variables.  353 

far  as  it  depends  upon  x  and  y.  Finally,  by  comparing  the 
total  differential  of  this  result  with  the  given  equation  we  de- 
termine dC  in  terms  of  z  and  dz,  and  thence  by  integration  the 
value  of  C. 

It  may  be  noticed  that  when  certain  terms  of  an  exact 
equation  forms  an  exact  differential,  the  remaining  terms  must 
also  be  exact.  It  follows  that  if  one  of  the  variables,  say  z 
can  be  completely  separated  from  the  other  two  (so  that  in 
equation  (i)  7?  becomes  a  function  of  z  only  and  Pand  Q  func- 
tions of  X  andy,  but  not  of  z)  the  terms  Pdx  -\-  Qdy  must  be 
thus  rendered  exact  if  the  equation  is  integrable."^  For  example^ 

zydx  —  zxdy  —  y'^dz  =  o. 
is  an  integrable  equation.    Accordingly,  dividing  hy  y^z,  which 
we  notice  separates  the  variable  z  from  x  and  y,  puts  it  in  the 

exact  form 

ydx  —  xdy      dz 

— ;;^    -  T  =  O' 

y  z 

of  which  the  integral  is  x  =  y  log  cz. 

Regarding  x,  y  and  z  as  coordinates  of  a  moving  pointy 
an  integrable  equation  restricts  the  point  to  motion  upon  one 
of  the  surfaces  belonging  to  the  system  of  surfaces  represented 
by  the  integral ;  in  other  words,  the  point  {x,  y,  z)  moves  in  an 
arbitrary  curve  drawn  on  such  a  surface.  Let  us  now  considef 
in  what  way  equation  (i)  restricts  the  motion  of  a  point  when 
it  is  not  integrable.  The  direction  cosines  of  a  moving  point 
are  proportional  to  dx,  dy,  and  dz;  hence,  denoting  them  by 
/,  m  and  n,  the  direction  of  motion  of  the  point  satisfying 
equation  (i)  must  satisfy  the  condition 

Pl+Qm+Rn  =  o.  (4) 

It  is  convenient  to  consider  in  this  connection  an  auxiliary 
system  of  lines  represented,  as  explained  in  Art.  12,  by  the 
simultaneous  equations 

dx  _dy  _dz  ,  . 

T~~Q~~R'  ^^• 

*  In  fact  for  this  case  the  condition  (3)  reduces  to  its  last  term,  which  ex» 
presses  the  exactness  of  Pdx  +  Qdy. 


354  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

The  direction  cosines  of  a  point  moving  in  one  of  the  hnes 
of  this  system  are  proportional  to  P,  Q  and  R.  Hence,  de- 
noting them  by  A,  yw,  Vy  equation  (4)  gives 

A/  -f  ^7n  -{-  vn  =  o  (6) 

for  the  relation  between  the  directions  of  two  moving  points, 
whose  paths  intersect,  subject  respectively  to  equation  (i)  and 
to  equations  (5).  The  paths  in  question  therefore  intersect  at 
right  angles;  therefore  equation  (i)  simply  restricts  a  point  to 
move  in  a  path  which  cuts  orthogonally  the  lines  of  the  auxili- 
ary system. 

Now,  if  there  be  a  system  of  surfaces  which  cut  the  auxiliary 
lines  orthogonally,  the  restriction  just  mentioned  is  completely 
expressed  by  the  requirement  that  the  line  shall  lie  on  one  of 
these  surfaces,  the  line  being  otherwise  entirely  arbitrary. 
This  is  the  case  in  which  equation  (i)  is  integrable."^ 

On  the  other  hand,  when  the  equation  is  not  integrable,  the 
restriction  can  only  be  expressed  by  two  equations  involving 
an  arbitrary  function.  Thus  if  we  assume  in  advance  one  such 
relation,  we  know  from  Art.  12  that  the  given  equation  (i) 
together  with  the  first  derivative  of  the  assumed  relation  forms 
a  system  admitting  of  solution  in  the  form  of  two  integrals* 
Both  of  these  integrals  will  involve  the  assumed  function.  For 
any  particular  value  of  that  function  we  have  a  system  of  lines 
satisfying  equation  (i),  and  the  arbitrary  character  of  the  func- 
tion makes  the  solution  sufficiently  general  to  include  all  lines 
which  satisfy  the  equation.f 

Prob.  76.  Show  that  the  equation 

{mz  —  ny)dx  -\-  {nx  —  lz)dy  +  (/f  —  mx)dz  =  o 
is  integrable,  and  infer  from  the  integral  the  character  of  the  auxil- 

*  It  follows  that,  with  respect  to  the  system  of  lines  represented  by  equations 
(5),  equation  (3)  is  the  condition  that  the  system  shall  admit  of  surfaces  cutting 
them  orthogonally.  The  lines  of  force  in  any  field  of  conservative  forces  form 
such  a  system,  the  orthogonal  surfaces  being  the  equipotential  surfaces. 

f  So  too  there  is  an  arbitrary  element  about  the  path  of  a  point  when  the 
single  equation  to  which  it  is  subject  is  integrable,  but  this  enters  only  into  one 
of  the  two  equations  necessary  to  define  the  path. 


Art  21.]  partial  equations,  first  order.  355 

iary  lines.     (Compare  the  illustrative  example  at  the  end  of  Art.  12.) 

(Ans.  nx  —  Iz  ^=  C(ny  —  mz).) 
Prob.  77.  Solve  ^^sV^  —  z'^dy  —  e^dz  =  o.     (Ans.  yz  =e''(i-{-cz).) 
Prob.  78.  Find  the  equation  which  in  connection  withj'  ='/(x) 
forms  the  solution  of  dz  =  aydx  -\-  bdy. 

Prob.  79.  Show  that  a  general  solution  of 
ydx  =  (x  —  z){dy  —  dz) 
is  given  by  the  equations 

y-  z=  (p{x),       y={x-  z)(p\x). 
(This  is  an  example  of  "  Monge's  Solution.") 


Art.  21.    Partial  Differential  Equations  of  First 
Order  and  Degree. 

Let  X  denote  an  unknown  function  of  the  two  independent 
variables  x  and  y,  and  let 

denote  its  partial  derivatives :  a  relation  between  one  or  both 
of  these  derivatives  and  the  variables  is  called  a  "  partial  dif- 
ferential equation  **  of  the  first  order.'  A  value  of  2:  in  terms  of 
jj:  and  y  which  with  its  derivatives  satisfies  the  equation,  or  a 
relation  between  jr,  y  and  z  which  makes  ^  implicitly  such  a 
function,  is  a  **  particular  integral."  The  most  general  equation 
of  this  kind  is  called  the  **  general  integral." 

If  only  one  of  the  derivatives,  say/,  occurs,  the  equation 
may  be  solved  as  an  ordinary  differential  equation.  For  if  j  is 
considered  as  a  constant,  /  becomes  the  ordinary  derivative  of 
^  with  respect  to  x;  therefore,  if  in  the  complete  integral  of 
the  equation  thus  regarded  we  replace  the  constant  of  integra- 
tion by  an  arbitrary  function  of  y,  we  shall  have  a  relation 
which  includes  all  particular  integrals  and  has  the  greatest  pos- 
sible generality.  It  will  be  found  that,  in  like  manner,  when 
both  p  and  g  are  present,  the  general  integral  involves  an  arbi- 
trary function. 

We  proceed  to  give  Lagrange's  solution   of  the  equation  of 


356  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

the  first  order  and  degree,  or  "  linear  equation,"  which  may  be 

written  in  the  form 

Pp+Qq  =  R,  (I) 

P,  Q  and  R  denoting  functions  of  x,  y  and  z.     Let  u  ^=  a,m 

which  ^^  is  a  function  of  x,  y  and  z,  and  a,  a  constant,  be  an 

integral  of  equation  (i).     Taking  derivatives  with  respect  to  x 

and  y  respectively,  we  have 

'du    ,92/  8«    ,   'du 

3^+9^^  =  °'        §7  +  ^^^^°' 

and  substitution  of  the  values  of  /  and  q  in  equation  (i)  gives 

the  symmetrical  relation 

j2)U       /du  du 

^a]^  + 097  +  ^5?=°-  (^> 

Consider  now  the  system  of  simultaneous  ordinary  differ- 
ential equations 

dx       dy      dz 

T^Q'^'R  *  (3> 

Let  «  =  o  be  one  of  the  integrals  (see  Art.  I2)  of  this  sys- 
tem.    Taking  its  total  differential, 

—-dx  +  :—dy  -\-  —-dz  =  o ; 

^x       ^   dy  d^ 

and  since  by  equations  (3)  dx,  dy  and  dz  are  proportional  to  P, 
Q  and  R,  we  obtain  by  substitution 

d^    ^dy^^d^ 
which  is  identical  with   equation   (2).     It  follows  that  every 
integral  of  the  system  (3)  satisfies  equation  (i),  and  conversely,, 
so  that  the  general  expression  for  the  integrals  of  (3)  will  be 
the  general  integral  of  equation  (i). 

Now  let  ^  =  ^  be  another  integral  of  equations  (3),  so  that 
V  is  also  a  function  which  satisfies  equation  (2).  As  explained 
in  Art.  12,  each  of  the  equations  u  =  a,  v  =  d  is  the  equation 
of  a  surface  passing  through  a  singly  infinite  system  of  lines 
belonging  to  the  doubly  infinite  system  represented  by  equa- 
tions (3).     What  we  require  is  the  general  expression  for  any 


Art.  21.] 


PARTIAL    EQUATIONS,    FIRST    ORDER. 


357 


surface  passing  through  lines  of  the  system  (and  intersecting 
none  of  them).  It  is  evident  that  f{u^  v)  =zf{a,  b)  =^  C  \s  such 
an  equation,"^  and  accordingly /(?/,  v),  where  /  is  an  arbitrary 
function,  will  be  found  to  satisfy  equation  (2).  Therefore,  to 
solve  equation  (i),  we  find  two  independent  integrals  u  z=z  a, 
V  =z  b  oi  the  auxiliary  system  (3),  (sometimes  called  Lagrange's 
equations,)  and  then  put 

u  =  4>{v),  (4) 

an  equation  which  is  evidently  equally  general  with/(?^,  v)  =  o. 
Conversely,  it  may  be  shown  that  any  equation  of  the  form 
(4),  regarded  as  a  primitive,  gives  rise  to  a  definite  partial 
differential  equation  of  Lagrange's  Hnear  form.  For,  taking 
partial  derivatives  with  respect  to  the  independent  variables 
X  and  y^  we  have 


'dy'^dz^ 


<P\v) 


<p\v) 


'%!-!/], 


.-dx 

'dv 


dz- 


I   9^ 


j^ 


and  eliminating  (f>'(v)  from  these  equations,  the  term  contain- 
ing/^ vanishes,  giving  the  result 


du  du 

du  du 

dudu 

dy  dz 
dy  dz 

P  + 

dz  dx 

d'v  dv 
dz  dx 

^  = 

dxdy 

dv  9^^ 

dxdy 

(5) 


which  is  of  the  form  Pp  -\-  Qq  —  R.\ 

*  Each  line  of  the  system  is  characterized  by  special  values  of  a  and  b  which 
we  may  call  its  coordinates,  and  the  surface  passes  through  those  lines  whose 
coordinates  are  connected  by  the  perfectly  arbitrary  relation /(«,  b)  =  C. 

f  These  values  of  P,  Q  and  R  are  known  as  the  "  Jacobians  "  of  the  pair 
of  functions  u,  v  with  respect  to  the  pairs  of  variables  y,  z\  z,  x  \  and  x,  y  re- 
spectively. Owing  to  their  analogy  to  the  derivatives  of  a  single  function  they 
are  sometimes  denoted  thus  : 


P  = 


9(«,  V) 


Q  = 


d(u,  v) 


R  = 


d(u,  v) 


d{y,  z)       "      9(2,  x)  d{x,  y) 

The  Jacobian  vanishes  if  the  functions  u  and  v  are  not  independent,  that  is 
to  say,  if  u  can  be  expressed  identically  as  a  function  of  v.     In  like  manner, 


358  DIFFERENTIAL    EQUATIONS.  [Chap.  VII» 

As  an  illustration,  let  the  given  partial  differential  equa- 
tion be 

(mz  —  ny)p  -\-  (tix  —  lz)q  z=  ly  —  mXf  (6) 

for  which  Lagrange's  Equations  are 

dx  dy  dz 

mz  —  ny~  nx  —  Iz  ^  ly  —  mx'  ^'^ 

These  equations  were  solved  at  the  end  of  Art.  12,  the  two 
integrals  there  found  being 

Ix -\- my -\- nz  =  a     and     x""  -\-  y""  -\- z"  =  b",  (8) 

Hence  in  this  case  the  system  of  *'  Lagrangean  lines  "  con- 
sists of  the  entire  system  of  circles  having  the  straight  line 

7=^=-  (9) 

I      m       n  ^^' 

for  axis.     The  general  integral  of  equation  (6)  is  then 

Ix -\- my  -\-  nz  =^  0(jr'  +y  +  ^^)^  (lo) 

which  represents  any  surface  passing  through  the  circles  just 
mentioned,  that  is,  any  surface  of  revolution  of  which  (9)  is  the 
axis.* 

Lagrange's  solution  extends  to  the  linear  equation  contain- 
ing n  independent  variables.     Thus  the  equation  being 

the  auxiliary  equations  are 

dx^      dx^  _         __  dx^  _  dz 

~ — '- — '- —  =  o  is  the  condition  that  (f>  (a  function  of  x,  y  and  2)  is  expressible 
0{x,  y\  2) 

identically  as  a  function  of  u  and  v,  that  is  to  say,  that  0  =  o  shall  be  an  in- 
tegral of  Pp-\-  Qq—  R. 

*  When  the    equation  Pdx  +  Qdy  -\-  Rdz  =  o  is   integrable  (as  it  is  in  the 
above  example;  see  Prob.  76,  An.  20),  its  integral,  which  may  be  put  in  the  form 

F=  C,  represents  a  singly  infinite  system  of  surfaces  which  the  Lagrangean 
lines  cut  orthogonally  ;  therefore,  in  this  case,  the  general  integral  may  be  de- 
fined as  the  general  equation  of  the  surfaces  which  cut  orthogonally  the  system 

K  =  C  Conversely,  starting  with  a  given  system  V  ^^  C,  u  =.  j\v)  is  the  gen- 
eral equation  of  the  orthogonal  surfaces,  if  u  =  a  and  v  =  d  are  integrals  of 


/  dx        ^  I  dv  I  hz 


Art.  22.]  COMPLETE    AND    GENERAL    INTEGRALS.  359 

and  if  u^  =  c^,  /^^  =  ^3,  ...//„  =  c„  are  independent   integrals, 
the  most  general  solution  is 

/(«!»  u^,  ,  .  ,  u^)  =  o, 
where /is  an  arbitrary  function. 

Prob.  80.    Solve  xz- V y^^  =  -V-     (Ahs.  xy—  0'  =/f— J.J 

Prob.  81.   Solve  (y  -j-  z)j>  -\-  {z  -\-  x)q  =1  x-\-  y, 
Prob.  82.  Solve  {x  -^ y){^p  —  q)  =  z. 

(Ans.   ^x-\-y)\ogz—x-=^f(x^y)) 
Prob.  83.  Solve  x{^y  —  z)p  -\- y{z  —  x)q  =  z{x  —  y). 

(Ans.  x-\-y-^  z  =  /{xyz).) 

Art.  22.    Complete  and  General  Integrals. 

We  have  seen  in  the  preceding  article  that  an  equation  be- 
tween three  variables  containing  an  arbitrary  function  gives 
rise  to  a  partial  differential  equation  of  the  linear  form.  It' 
follows  that,  when  the  equation  is  not  linear  in/  and  g,  the 
general  integral  cannot  be  expressed  by  a  single  equation  of 
the  form  cp{Uy  7j)  =:  o;  it  will,  however,  still  be  found  to  depend 
upon  a  single  arbitrary  function. 

It  therefore  becomes  necessary  to  consider  an  integral  hav- 
ing as  much  generality  as  can  be  given  by  the  presence  of  arbi- 
trary constants.  Such  an  equation  is  called  a  "  complete  in- 
tegral "  ;  it  contains  two  arbitrary  constants  (n  arbitrary  con- 
stants in  the  general  case  of  n  independent  variables),  because 
this  is  the  number  which  can  be  eliminated  from  such  an  equa- 
tion, considered  as  a  primitive,  and  its  two  derived  equations. 
For  example,  if 

(x-ay  +  (y-dy  +  ^'  =  k\ 

a  and  b  being  regarded  as  arbitrary,  be  taken  as  the  primitive, 
the  derived  equations  are 

X  —  a  ^  zp  =z  o^         y  —  b  -\-  zq  ^  O, 
and  the  elimination  of  a  and  b  gives  the  differential  equation 

^X/  +  ?'+!)  =  '^'. 

of  which  therefore  the  given  equation  is  a  complete  integral. 


360  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

Ge9metncally,  the  complete  integral  represents  a  doubly  in- 
finite system  of  surfaces  ;  in  this  case  they  are  spherical  sur- 
faces having  a  given  radius  and  centers  in  the  plane  of  xy. 

In  general,  a  partial  differential  equation  of  the  first  order 
with  two  independent  variables  is  of  the  form 

F{x,y,z,p,q)  =  0,  (I) 

and  a  complete  integral  is  of  the  form 

f{x,  y,  z,  a,  b)  =  o.  (2) 

In  equation  (i)  suppose  x,  y  and  z  to  have  special  values, 
namely,  the  coordinates  of  a  special  point  A  ;  the  equation 
becomes  a  relation  between  p  and  q.  Now  consider  any  sur- 
face passing  through  A  of  which  the  equation  is  an  integral  of 
(i),  or,  as  we  may  call  it,  a  given  "integral  surface  "  passing 
through  A.  The  tangent  plane  to  this  surface  at  A  determines 
values  of  /  and  q  which  must  satisfy  the  relation  just  men- 
tioned. Consider  also  those  of  the  complete  integral  surfaces 
[equation  (2)]  which  pass  through  A.  They  form  a  singly  in- 
finite system  whose  tangent  planes  at  A  have  values  of  p  and 
^  which  also  satisfy  the  relation.  There  is  obviously  among 
them  one  which  has  the  same  value  of  /,  and  therefore  also 
the  same  value  of  q,  as  the  given  integral.  Thus  there  is  one 
of  the  complete  integral  surfaces  which  touches  at  A  the  given 
integral  surface.  It  follows  that  every  integral  surface  (not  in- 
cluded in  the  complete  integral)  must  at  every  one  of  its  points 
touch  a  surface  included  in  the  complete  integral.* 

It  is  hence  evident  that  every  integral  surface  is  the  en- 
velope of  a  singly  infinite  system  selected  from  the  complete 
integral  system.  Thus,  in  the  example  at  the  beginning  of 
this  article,  a  right  cylinder  whose  radius  is  k  and  whose  axis 
lies  in  the  plane  of  xy  is  an  integral,  because  it  is  the  envelope 

*  Values  of  x,  y,  and  z,  determining  a  point,  together  with  values  of/  and  q, 
determining  the  direction  of  a  surface  at  that  point,  are  said  to  constitute  an 
"element  of  surface."  The  theorem  shows  that  the  complete  integral  is  "com- 
plete "  in  the  sense  of  including  all  the  surface  elements  which  satisfy  the  differ- 
•ential  equation.  The  method  of  grouping  the  "consecutive"  elements  to  form 
an  integral  surface  is  to  a  certain  extent  arbitrary. 


Art.  22.]  complete  and  general  integrals.  361 

of  those  among  the  spheres  represented  by  the  complete  in- 
tegral whose  centers  are  on  the  axis  of  the  cylinder.  If  we 
make  the  center  of  the  sphere  describe  an  arbitrary  curve  in 
the  plane  of  xy  we  shall  have  the  general  integral  in  this  ex- 
ample. 

In  general,  if  in  equation  (2)  an  arbitrary  relation  between 
a  and  b,  such  as  <^  =  0(^),  be  established,  the  envelope  of  the 
singly  infinite  system  of  surfaces  thus  defined  will  represent 
the  general  integral.  By  the  usual  process,  the  equation  of 
the  envelope  is  the  result  of  eliminating  a  between  the  two 
equations 

f{x,y,  z,  a,  (f){a))  =  o,     ^/(^,  fy  ^,  a,  (p{a) )  =  o.         (3) 

These  two  equations  together  determine  a  line,  namely,  the 
"  ultimate  intersection  of  two  consecutive  surfaces."  Such 
lines  are  called  the  "  characteristics  "  of  the  differential  equa- 
tion. They  are  independent  of  any  particular  form  of  the 
complete  integral,  being  in  fact  lines  along  which  all  integral 
surfaces  which  pass  through  them  touch  one  another.  In  the 
illustrative  ex.mple  above  they  are  equal  circles  with  centers 
in  the  plane  of  xy  and  planes  perpendicular  to  it."^ 

The  example  also  furnishes  an  instance  of  a  **  singular  so- 
lution "  analogous  to  those  of  ordinary  differential  equations. 

*The  characteristics  are  to  be  regarded  not  merely  as  lines,  but  as  "  linear 
elements  of  surface,"  since  they  determine  at  each  of  their  points  the  direction 
of  the  surfaces  passing  through  them,  Thus,  in  the  illustration,  they  are  cir- 
cles regarded  as  great-circle  elements  of  a  sphere,  or  as  elements  of  a  right 
cylinder,  and  may  be  likened  to  narrow  hoops.  They  constitute  in  all  cases  a 
triply  infinite  system.  The  surfaces  of  a  complete  integral  system  contain  them 
all,  but  they  are  differently  grouped  in  different  integral  surfaces. 

If  we  arbitrarily  select  a  curve  in  space  there  will  in  general  be  at  each  of 
its  points  but  one  characteristic  through  which  the  selected  curve  passes;  that 
is.  whose  tangent  plane  contains  the  tangent  to  the  selected  curve.  These  char- 
acteristics (for  all  points  of  the  curve)  form  an  integral  surface  passing  through 
the  selected  curve  ;  and  it  is  the  only  one  which  passes  through  it  unless  it  be 
itself  a  characteristic.  Integral  surfaces  of  a  special  kind  result  when  the  se- 
lected curve  is  reduced  to  a  point.  In  the  illustration  these  are  the  results  of 
rotating  the  circle  about  a  line  parallel  to  the  axis  of  z. 


36;^  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

For  the  planes  z  =^  :t  k  envelop  the  whole  system  of  spheres 
represented  by  the  complete  integral,  and  indeed  all  the  sur- 
faces included  in  the  general  integral.  When  a  singular  solu- 
tion exists  it  is  included  in  the  result  of  eliminating  a  and  b 
from  equation  (2)  and  its  derivatives  with  respect  to  a  and  b^ 
that  is,  from 

but,  as  in  the  case  of  ordinary  equations,  this  result  may  in- 
clude relations  which  are  not  solutions. 

Prob.  84.  Derive  a  differential  equation  from  the  primitive 
Ix  +  my  -\-  nz  =  a^  where  /,  m,  n  are  connected    by   the    relation 

Prob.  85.  Show  that  the  singular  solution  of  the  equation, 
found  in  Prob.  84  represents  a  sphere,  that  the  characteristics  con- 
sist of  all  the  straight  lines  which  touch  this  sphere,  and  that  the 
general  integral  therefore  represents  all  developable  surfaces  which 
touch  the  sphere. 

Prob.  86.  Find  the  integral  which  results  from  taking  in  the 
general  integral  above  /'  -\-m^  =  cos'  6^  (a  constant)  for  the  arbitrary 
relation  between  the  parameters. 

Art,  23.     Complete  Integral  for  Special  Forms. 

A  complete  integral  of  the  partial  differential  equation 

F{x,y,  z,p,  q)  =  0  (i) 

contains  two  constants,  a  and  b.  If  a  be  regarded  as  fixed  and 
b  as  an  arbitrary  parameter,  it  is  the  equation  of  a  singly  in- 
finite system  of  surfaces,  of  which  one  can  be  found  passing 
through  any  given  point.  The  ordinary  differential  equation 
of  this  system,  which  will  be  independent  of  b,  may  be  put  in 

the  form 

dz  =  pdx  +  qdy,  (2)- 

in  which  the  coefficients/  and  q  are  functions  of  the  variables 
and  the  constant  a.  Now  the  form  of  equation  (2)  shows  that 
these  quantities  are  the  partial  derivatives  of  z,  in  an  integral 
of  equation  (i);  therefore  they  are  values  of/  and  q  which 


Art.  23.]        complete  integral  for  special  forms.  365 

satisfy  equation  (i).  Conversely,  if  values  oi  p  and  q  in  terms 
of  the  variables  and  a  constant  a  which  satisfy  equation  (i)  are 
such  as  to  make  equation  (2)  the  differential  equation  of  a  sys- 
tem of  surfaces,  these  surfaces  will  be  integrals.  In  other 
words,  if  we  can  find  values  of/  and  q  containing  a  constant  a. 
which  satisfy  equation  (i)  and  make  dz  =^  pdx -\- qdy  inte- 
grable,  we  can  obtain  by  direct  integration  a  complete  inte^ 
gral,  the  integration  introducing  a  second  constant. 

There  are  certain  forms  of  equations  for  which  such  values 
of  p  and  q  are  easily  found.  In  particular  there  are  forms  in 
which  p  and  q  admit  of  constant  values,  and  these  obviously 
make  equation  (2)  integrable.  Thus,  if  the  equation  contains 
p  and  q  only,  being  of  the  form 

^(/,  ^)  =  o,  (3> 

we  may  put/  =  a  and  q  =  b,  provided 

F(a,d)  =  o.  (4> 

Equation  (2)  thus  becomes 

djs  —  adx  +  bdy^ 
whence  we  have  the  complete  integral 

z  =  ax-\-by-^c,  (5) 

in  which  a  and  b  are  connected  by  the  relation  (4)  so  that  ^,  b 
and  c  are  equivalent  to  two  arbitrary  constants. 

In  the  next  place,  if  the  equation  is  of  the  form 

z=px-\-qy^f{p,q\  (6) 

which  is  analogous  to  Clairaut's  form,  Art.  10,  constant  values 
of/  and  q  are  again  admissible  if  they  satisfy 

z^ax  +  by-\-f(a,b\  (7) 

and  this  is  itself  the  complete  integral.  For  this  equation  is 
of  the  form  z  =ax  -\-  by  -\-  c,  and  expresses  in  itself  the  rela- 
tions between  the  three  constants.  Problem  84  of  the  preced- 
ing article  is  an  example  of  this  form. 

In  the  third  place,  suppose  the  equation  to  be  of  the  form 
F{z,p,q)  =  o,  (8) 


364  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

in  which  neither  x  nox  y  appears  expHcitly.  If  we  assume 
^  =:  apy  p  will  be  a  function  of  z  determined  from 

F{z,  p,  ap)  =  o,     say    /  =  0(4  (9) 

Then  dz  =pdx  +  qdy  —  o  becomes  dz  =  (f)(z){dx  -j-  ady),  which  is 
integrable,  giving  the  complete  integral 

^+''->'  =  /^)+^-  ^■°) 

A  fourth  case  is  that  in  which,  while  z  does  not  explicitly 
occur,  it  is  possible  to  separate  x  and/  from  y  and  q,  thus  put- 
ting the  equation  in  the  form 

/.(^>/)  =/>(/,  ?)•  (II) 

If  we  assume  each  member  of  this  equation  equal  to  a  con- 
stant Uy  we  may  determine/  and  q  in  the  forms 

/  =  ^ix,  a),  q  =  cp^y,  a).  (12) 

and  dz  =  pdx  +  qdy  takes  an  integrable  form  giving 

z  =  J  cpXx,d)dx  + J*cf),{y,d)dy-\-b.  (13) 

It  is  frequently  possible  to  reduce  a  given  equation  by  trans- 
formation of  the  variables  to  one  of  the  four  forms  considered 
in  this  article.*  For  example,  the  equation  x'^p^'  -\- y^  q^  z=  z^ 
may  be  written 

{xdz\  ,    {ydz\_ 

Kzdxl'^  \zdyl  ~  ^' 

*The  general  method,  due  to  Charpit.  of  finding  a  proper  value  oip  consists 
of  establishing,  by  means  of  the  condition  of  integrability,  a  linear  partial  dif- 
ferential equation  for/,  of  which  we  need  only  a  particular  integral.  This  may 
be  any  value  of  p  taken  from  the  auxiliary  equations  employed  in  Lagrange's 
process.  See  Boole,  Differential  Equations  (London  1865),  p.  336  ;  also  For- 
syth, Differential  Equations  (London  1885),  p.  316,  in  which  the  auxiliary  equa- 
tions are  deduced  in  a  more  general  and  symmetrical  form,  involving  both  / 
and  q.  These  equations  are  in  fact  the  equations  of  the  characteristics  regarded 
as  in  the  concluding  note  to  the  preceding  article.  Denoting  the  partial  deriva- 
tives of  F{x,  y,  z,  p,  q)  by  X,  Y,  Z,  P,  Q,  they  are 

dx  dy  dz  dp  dq 

~P'=~Q-    Pp+Qq  =~  ^=h^  =~    Y+Zq 

See  Jordan's  Course  d' Analyse  (Paris,  1887),  vol.  in,  p.  318  ;  Johnson's  Differ- 
ential Equations  (New  York,  1889),  p.  300.  Any  relation  involving  one  or  both 
the  quantities  /  and  y,  combined   with  P=o,  will  furnish  proper  values  ot  p 


Art.  -^4.]  PARTIAL    EQUATIONS,    SECOND    ORDER.  365 

whence,  putting  x'  =  log  x^  y'  =  log/,  z'  =  log  ^,  it  becomes 
p'""  -\-q'''  =  I,  which  is  of  the  form  F{^p\  q')  =  o,  equation  (3). 
Hence  the  integral  is  given  by  equation  (5)  when  a^  -\-  b*  =z  i; 
it  may  therefore  be  written 

z'  =  x'  cos  a  -\-  y'  sin  a-\-  c, 
and  restoring  x,  y,  and  z,  that  of  the  given  equation  is 

Prob.  87.  Find  a  complete  integral  for/'  —  q^  z=.  \, 

(Ans.  2  =  a:  sec  a  -^-y  tan  a  -j-  b?) 
Prob.  88.  Find  the  singular  solution  oi  z  =  px  -\-  qy  -\-  pq. 

(Ans.  2  =  —  ^^. ) 
Prob.  89.  Solve  by  transformation  q  =  2yp'. 

(Ans.  2^  =  d!^  -f  dty  +  <^.) 
Prob.  90.  Solve  z(p'^—q^)  z=  x  —  y. 

(Ans.  2^  =(x  +  a)^  +  (7  +  «)^  +  ^.) 
Prob.  91.  Show  that  the  solution  given  for  the  form  J^(z,p,  q)  =  o 
represents   cylindrical  surfaces,  and  that  J^{z,  o,  o)  =  o  is  a  singular 
solution. 

Prob.  92.  Deduce  by  the  method  quoted  in  the  foot-note  two 
complete  integrals  of  pq  =  px  -\-  qy. 

(Ans.   2z=(-^-\-  oiyj  -|-/J,  3indz  =  xy+y  i/{x'  +  «)  +  b.) 

Art.  24.    Partial  Equations  of  Second  Order. 

We  have  seen  in  the  preceding  articles  that  the  general 
solution  of  a  partial  differential  equation  of  the  first  order  de- 
pends upon  an  arbitrary  function  ;  although  it  is  only  when 
the  equation  is  linear  in  /  and  q  that  it  is  expressible  by  a 
single  equation.  But  in  the  case  of  higher  orders  no  general 
account  can  be  given  of  the  nature  of  a  solution.  Moreover, 
when  we  consider  the  equations  derivable  from  a  primitive  con- 
taining arbitrary  functions,  there  is  no  correspondence  between 
their  number  and  the  order  of  the  equation.     For  example,  if 

and  ^.     Sometimes  several  such  relations  are  readily  found  ;  for  example,  for 
the  equation  z-=pq  we  thus  obtain  the  two  complete  integrals 

z^(,y^a){x^b)     and     42  =^f  +  O)/ + /jV. 


36€  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

the  primitive  with  two  independent  variables  contains  two  ar- 
bitrary functions,  it  is  not  generally  possible  to  eliminate  them 
and  their  derivatives  from  the  primitive  and  its  two  derived 
•equations  of  the  first  and  three  of  the  second  order. 

Instead  of  a  primitive  containing  two  arbitrary  functions, 
let  us  take  an  equation  of  the  first  order  containing  a  single 
arbitrary  function.  This  may  be  put  in  the  form  u  =  (p{z>\ 
u  and  V  now  denoting  known  functions  of  jf,  j,  ^,  ^,and  q. 
cf)\v)  may  now  be  eliminated  from  the  two  derived  equations 
as  in  Art.  21.     Denoting  the  second  derivatives  of  z  by 

''~dx''         '--dx-dy'         ^-Qy 
the  result  is  found  to  be  of  the  form 

RrJ^Ss-^Tt^  U{rt  -  s')  =  F",  (i) 

in  which  R,  S,  T,  U,  and  V  are  functions  of  x,  y,  z,  p,  and  q. 
With  reference  to  the  differential  equation  of  the  second  order 
the  equation  u  =  (p{v)  is  called  an  "  intermediate  equation  of 
the  first  order  " :  it  is  analogous  to  the  first  integral  of  an  ordi- 
nary equation  of  the  second  order.  It  follows  that  an  inter- 
mediate equation  cannot  exist  unless  the  equation  is  of  the 
iform  (i);  moreover,  there  are  two  other  conditions  which 
must  exist  between  the  functions  R,  5,  7",  and  U, 

In  some  simple  cases  an  intermediate  equation  can  be  ob- 
tained by  direct  integration.  Thus,  if  the  equation  contains 
■derivatives  with  respect  to  one  only  of  the  variables,  it  may  be 
treated  as  an  ordinary  differential  equation  of  the  second  order, 
the  constants  being  replaced  by  arbitrary  functions  of  the 
other  variable.  Given,  for  example,  the  equation  xr  — /  =  xy^ 
which  may  be  written 

xdp  —  pdx  =  xy  dx. 

This  becomes  exact  with  reference  to  x  when  divided  by  x^^ 
and  gives  the  intermediate  equation 

p=^yx\ogx-\-x(l}{y\ 
A  second  integration  (and  change  in  the  form  of  the  arbitrary 
function)  gives  the  general  integral 

z  =  iyx'  log  X  4-  x'(p{y)  +  tp(y). 


Art.  34.]  partial  equations,  second  order.  367 

Again,  the  equation /  + r +-y  =  i  is  already  exact,  and 
gives  the  intermediate  equation 

which  is  of  Lagrange's  form.     The  auxiHary  equations*  are 

of  which  the  first  gives  x  —  y  =:^  a,  and  ehminating  x  from  the 
second,  its  integral  is  of  the  form 

z  =  a-]-  <p(y)  -\-e~yb. 

Hence,  putting  b  =  ipia),  we  have  for  the  final  integral 

z  =  x-{-  (p{y)  +  e-ytp{x  —y), 

in  which  a  further  change  is  made  in  the  form  of  the  arbitrary 
function  0. 

Prob,  93.  Solve  /  —  ^  =  ^*  +  ^^• 

(Ans.  z  =y(e^  —  <?*)  +  0(^)  +  g^fp{x).) 

Prob.  94.  Solve  r  +/  =/. 

(Ans.  z  =  logie'^ytpiy)  -  ^"^^J  +  f{y).) 

Prob.  95.  Solve  y(j  —  i)  =  x. 

(Ans.  z=  {x  -]-y)  \ogy  +  0(:r)  -U  tp{x   \  y).) 

Prob.  96.  Solve /J  —  ^r  =  o.  (Ans.  x  —  (p(y)  -\-  ^'(s).) 

Prob.  97.  Show  that  Monge's  equations  (see  foot-note)  give  for 
Prob.  96  the  intermediate  integral  /  =  <p(z)  and  hence  derive  the 
solution. 

*  In  Monge's  method  (for  which  the  reader  must  be  referred  to  the  complete 
treatises)  of  finding  an  intermediate  integral  of 

J^r -\-  Ss -{-  Tt  =  V 
when  one  exists,  the  auxiliary  equations 

/?c/f  -  Sdy  dx  +  Tdx^  =  o,         Rdp  dy  -f-  Tdq  dx  =  Vdx  dy 
are  established.     These,  in  connection  with 

dz  =  pdx  -\-  qdy, 
form  an  incomplete  system  of  ordinary  differential  equations,  between  the  five 
variables  x,  y,  z,  p,  and  q.  But  if  it  is  possible  to  obtain  two  integrals  of  the 
system  in  the  form  u  =  a,  v  =  d,  u  =  <p{v)  will  be  the  intermediate  integral. 
The  first  of  the  auxiliary  equations  is  a  quadratic  giving  two  values  for  the  ratio 
dy.dx.  If  these  are  distinct,  and  an  intermediate  integral  can  be  found,  for 
€ach,  the  values  of  p  and  q  determined  from  them  will  make  dz  —pdx  -{-qdy 
jntegrable,  and  give  the  general  integral  at  once. 


368  DIFFERENTIAL    EQUATIONS.  [Chap.  VIL 

Prob.  98.  Derive  by  Monge's  method  for  q'*r  —  2pqs -\- p^t  =  o 
the  intermediate  integral/  =  q  <p{z)f  and  thence  the  general  integral. 

(Ans.  y  +  xc/){z)  =  tp{z),) 

Art.  25.    Linear  Partial  Differential  Equations. 

Equations  which  are  linear  with  respect  to  the  dependent 
variable  and  its  partial  derivatives  may  be  treated  by  a  method 
analogous  to  that  employed  in  the  case  of  ordinary  differential 
equations.  We  shall  consider  only  the  case  of  two  independ- 
ent variables  x  and  /,  and  put 

so  that  the  higher  derivatives  are  denoted  by  the  symbols  D^, 
DD\  D'^^  D^,  etc.  Supposing  further  that  the  coefficients  are 
constants,  the  equation  may  be  written  in  the  form 

f{D,  D')z  =  F{x,  y),  (I) 

in  which  /  denotes  an   algebraic   function,   or  polynomial,  of 

which  the  degree  corresponds  to  the  order  of  the  differential 

equation.     Understanding  by  an  "integral"  of  this  equation 

an  explicit  value  of  z  in  terms  of  x  and  y,  it  is  obvious,  as  in 

Art.  15,  that  the  sum  of  a  particular  integral  and  the  general 

integral  of 

AD,  D')z  =  o  (2> 

will  constitute  an  equally  general  solution  of  equation  (i).  It 
is,  however,  only  when  /{£>,  D')  is  a  homogeneous  function  of  D 
and  D'  that  we  can  obtain  a  solution  of  equation  (2)  containing 
n  arbitrary  functions,*  which  solution  is  also  the  "  comple- 
mentary function"  for  equation  (i). 

Suppose  then  the  equation  to  be  of  the  form 
d^z  d^z  d^z 

^•^  +  ^-^=-^+---+^-^"  =  °'  (3> 

and  let  us  assume  z  =  (p{y  -\-  mx),  (4) 

*  It  is  assumed  that  such  a  solution  constitutes  the  general  integral  of  an 
equation  of  the  «th  order;  for  a  primitive  containing  more  than  n  independent 
arbitrary  functions  cannot  give  rise  by  their  elimination  to  an  equation  of  the 
«th  order. 


Art.  25.]  LINEAR  partial  equations.  369 

where  m  is  a.  constant  to  be  determined.  From  equation  (4), 
£}b  =  m(p\y  +  'ynx)  and  D'z  =  4>\y  -\-  mx),  so  that  Dz  =  mU z^ 
D^z  =  tn^D'^Zy  DD'z  =  mD^y,  etc.  Substituting  in  equation  (3) 
and  rejecting  the  factor  D^*^z  or  <p^''\y  -["  ^^)>  we  have 

A^m-  +  A,M--'  +  ,,,  +  A„  =  o  (5) 

for  the  determination  of  m.  li  m^,  m^j ,  ,  ,  m^  are  distinct  roots 
of  this  equation, 

^  =   4>iy  +  ^,^)  +0,(j^  +  ^a^)  +  •   •   •  +  (t>n{y  +  mnX)     (6) 

is  the  general  integral  of  equation  (3). 

(Pz      d^z 
For  example,  the  general  integral  of  -71  —  -y-a  =  o  is  thus 

found  to  be  ^  =  0(^  +  ■^)  +  ^{y  ~  ^)'  Any  expression  of  the 
form  Axy  +  Bx  -\-  Cy  -{-  D  \s  2i  particular  integral ;  accordingly 
it  is  expressible  as  the  sum  of  certain  functions  of  x -{- y  and 
X  —  y  respectively. 

The  homogeneous  equation  (3)  may  now  be  written  sym- 
bolically in  the  form 

{D  -  m,D'){D  -  mfi') .  .  .  (Z>  -  m^D')z  =  o,  (7) 

in  which  the  several  factors  correspond  to  the  several  terms  of 
the  general  integral.  If  two  of  the  roots  of  equation  (5)  are 
equal,  say,  to  w,,  the  corresponding  terms  in  equation  (6)  are 
equivalent  to  a  single  arbitrary  function.  To  form  the  general 
integral  we  need  an  integral  of 

{B  -  m.DJz  =  o  (8) 

in  addition  to  (f>{y  +  m^x).     This  will  in  fact  be  the  solution  of 

{D  —  mfi')z  =  <p(y  +  m,x);  (9) 

for,  if  we  operate  with  D  —  m^D'  upon  both  members  of  this 

equation,  we  obtain  equation  (8).     Writing  equation  (9)  in  the 

form 

p  —  m,g=  (p{y  +  mx), 

Lagrange's  equations  are 

dy  dz 

dx  — = 


giving  the  integrals  y  +  ^1^  =  a^  z  =  x(p{a)  +  b.     Hence  the 
integral  of  equation  (9)  is 

z  =  x(l){y  -\-  m^x)  -\-  tp(y  -\-  m^x),  (10)  . 


370  DIFFERENTIAL    EQUATIONS.  [Chap.  VII. 

and  regarding  0  also  as  arbitrary,  these  are  the  two  independ- 
ent terms  corresponding  to  the  pair  of  equal  roots. 

If  equation  (5)  has  a  pair  of  imaginary  roots  m  =1  fA,  :^  iv, 
the  corresponding  terms  of  the  integral  take  the  form 

(p{y  -\-fAX  -\-ivx)  -^  '(p(y  -\-  ^x  —  irx), 

which  when  0  and  tp  are  real  functions  contain  imaginary 
terms.  If  we  restrict  ourselves  to  real  integrals  we  cannot 
now  say  that  there  are  two  radically  distinct  classes  of  inte- 
grals ;  but  if  any  real  function  oi  y  -\-  ^x  ~\-  ivx  be  put  in  the 
form  X-\-iY,  either  of  the  real  functions  X  or  Fwill  be  an 
integral  of  the  equation.     Given,  for  example,  the  equation 

dx^'^d/"'"' 
of  which  the  general  integral  is 

2  =  (p[y  -|-  tx)  +  tp{y  —  ix)  ; 

to  obtain  a  real  integral  take  either  the  real  or  the  coefficient 
of  the  imaginary  part  of  any  real  form  of  cf){y  +  ix).  Thus,  if 
(pit)  r=  ^  we  find  e^  cos;i;  and  e^  smx,  each  of  which  is  an 
integral. 

As  in  the  corresponding  case  of  ordinary  equations,  the 
particular  integral  of  equation  (i)  may  be  made  to  depend 
upon  the  solution  of  linear  equations  of  the  first  order.     The 

inverse  symbol  jz jr,F(x^  y)  in  the  equation  corresponding 

to  equation  (14),  Art.  16,  denotes  the  value  of  z  in 

{D  —  mD')z  =-  F{x,  y)     ox    p  —  mq  =  Fix,  y),  (ii) 

For  this  equation  Lagrange's  auxiliary  equations  give 
y  +  mx  —  a,      z  —  i  Fix,  a  —  mx)dx  -{-  b  =  F^{x,  a)  +  b, 

and  the  general  integral  is 

z  =  F^{x,y  -\-  mx)  -\-  <p(y  -\-  mx),  (12) 

The  first  term,  which  is  the  particular  integral,  may  there- 
fore  be  found   by   subtracting  mx  from  y   in   F{x,  y),  inte- 


Art.  25.]  linear  partial  equations.  ,  371 

grating  with  respect  to  x,  and  then  adding  mx  to  y  in  the 
result.* 

For  certain  forms  of  F{x^  y)  there  exist  more  'expeditious 
methods,  of  which  we  shall  here  only  notice  that  which  appHes 
to  the  form  F{ax  -\-  by).  Since  DF{ax  -\-  by)  =  aF\ax  +  by) 
and  D' F{ax  -[-  by)  =  bF\ax  -\-  by),  it  is  readily  inferred  that, 
when/(Z?,  D')  is  a  homogeneous  function  of  the  ;^th  degree, 

/(A  n)F(,ax  +  by)  =f{a,  b)F^'\ax  +  by).  (13) 

That  is,  if  t  ^=  ax  -{-  by,  the  operation  of  /(A  D')  on  F{()  is 
equivalent  to  multiplication  hy  f{a,b)  and  taking  the  nXh  de- 
rivative, the  final  result  being  still  a  function  of  t.  It  follows 
that,  conversely,  the  operation  of  the  inverse  symbol  upon  a 
function  of  /  is  equivalent  to  dividing  by /(^,  b)  and  integrating 
n  times.     Thus, 

7(^)  ^("^^ + ^-^)  =f^SS-  ■  •/^(')'^^-    ('4) 

When  ax  -\-by  \?>  2l  multiple  oi  y  -\-  m^x,  where  w^  is  a  root  of 
equation  (5),  this  method  fails  with  respect  to  the  correspond- 
ing symbolic  factor,  giving  rise  to  an  equation  of  the  form  (9), 
of  which  the  solution  is  given  in  equation  (10).  Given,  for  ex- 
ample, the  equation 

Ix^-^d^y-^  ^'  =  ^^"  (^-7)+  sin(^+7) 
or        {D  —  D')  {D  +  2D')z  =  sin  (^  —  7)  +  sin  {x  +  y). 
The  complementary  function  is  (f){y  -\-  x)  -\-  ip{y  —  2x).     The 
part  of  the  particular  integral  arising  from  sin  {x  —  y),  in  which 

az=i,b  =  —i,\s /    /sin  tdf  =  ~  sin  {x  —  y).     That  aris- 

*  The  symbolic  form  of  this  theorem  is 

n  -mD'^^'''  y^  =  e^^D'Je-  mxD'pi^x,  y)dx 

corresponding  to  equation  (13),  Art.  16.  The  symbol  e^'^D'  here  indicates  the 
addition  of  mx  to  y  in  the  operand.  Accordingly,  using  the  expanded  form 
of  the  symbol, 


^-D>F{y)  =  (I  +  ^^  -  +  -;rr  :ir2  + . . .)  Fh)  =  Hy  +  mx). 


Jy  ~^  TT  dy 
ihe  symbolic  expression  of  Taylor's  Theorem. 


372  DIFFERENTIAL    EQUATIONS.  [ChAP.  VIL 

ing  from  sin  (;tr  +  j)  which  is  of  the  form  of  a  term  in  the  com- 
plementary; function  is jz yr,  cos  {x  -\-j),  which  by  equa- 
tion (lo)  is  —  -J;!:  cos  (-^+7).  Hence  the  general  integral  of 
the  given  equation  is 

^  =  0(J  +  ^)  +  iij  -  2^)  +  i  sin  {x-J^)-  i^  cos  {x  +  j). 
If  in  the  equation /(A  B');s  =  o  the  symbol /(A  D'),  though 
not  homogeneous  with  respect  to  D  and  B\  can  be  separated 
into  factors,  the  integral  is  still  the  sum  of  those  corresponding 
to  the  several  symbolic  factors.  The  integral  of  a  factor  of 
the  first  degree  is  found  by  Lagrange's  process ;  thus  that  of 

{p-mD'-d)z  =  o  (15) 

is  2  =  e''''cp{y  +  mx).  (16) 

But  in  the  general  case  it  is  not  possible  to  express  the 
solution  in  a  form  involving  arbitrary  functions.  Let  us,  how- 
ever, assume 

^=^^*+*^,  (17) 

where  c,  h,  and  k  are  constants.  Since  Z>^*  +  *^  =  ^^*  +  *^ 
and  i)V^+*^= /^^^  +  *^  substitution  in  f'{D,  D')z  =  o  gives 
cf{h,  k^e^''^^^  =  o.  Hence  we  have  a  solution  of  the  form  (17) 
whenever  k  and  ^  satisfy  the  relation 

/(^,  ^)  =  o,  (18) 

c  being  altogether  arbitrary.  It  is  obvious  that  we  may  also 
write  the  more  general  solution 

^  =  :2'^^"+^w^  (19) 

where  k  =  F(Ji)  is  derived  from  equation  (18),  and  c  and  h  admit 
of  an  infinite  variety  of  arbitrary  values. 

Again,  since  the  difference  of  any  two  terms  of  the  form 
^x-\-F{h)y  ^ith  different  values  of  h  is  included  in  expression 
(19),  we  infer  that  the  derivative  of  this  expression  with  respect 
to  h  is  also  an  integral,  and  in  like  manner  the  second  and 
higher  derivatives  are  integrals. 
For  example,  if  the  equation  is 

d'^z      dz 

d?~~d~y^^' 


Art.  25.]  linear  partial  equations.  373 

for  which  equation   (i8)  is  J^  —  /^  =  o,  we  have  classes  of  in- 
tegrals of  the  forms 

^x  +  Kh^(^^  _^  2hyf  +  2y)\       ^"  +  ''\{x  +  2hyY  +  6;/(;tr  +  2hy)\ 


In  particular,  putting  /^  =  o  we  obtain  thx>  algebraic  integrals 
c^x,  c^(x'-\-2j/),  c,(x'-\-6xy),  etc. 

The  solution  of  a  linear  partial  differential  equation  with 
variable  coefficients  may  sometimes  be  effected  by  a  change  of 
the  independent  variables  as  illustrated  in  some  of  the  exam- 
ples below. 

Prob.  99.  Show  that  if  m^  is  a  triple  root  the  corresponding 
terms  of  the  integral  are  ^VC.)'  +  ^ix)  -\-  x(p(y-\-  m^x)-{-  x{y-\-^h^)' 

c  1         9'^            9"^  9^^ 

Prob.  100.  Solve  2^-^  —  3;^-^ 2^-2  =  o. 

Prob.  loi.  Solve  ^-^  +  2^^  +  9?  =  ^• 

(Ans.  2:  =  ^{x)  ■\-  tp(x  -{- y)  -\-  xx(x  -i-y)  —y  log  x.) 
Prob.  102.  Solve  (Z>^  +  5Z>Z>'  +  6Z>")s  =  (j'  -  2x)-\ 

(Ans.  2?  =  0(^  —  2x)  +  ^(jj^  —  3jc)  +  -^  log  (j^'  —  2^).) 

Prob.  103.  Solve  — ^  —  ——-  +  :r z  =  o. 

9^       9^t9^      9j 

Prob.  104.  Show  that  for  an  equation  of  the  form  (15)  the  solu- 
tion given  by  equation  (19)  is  equivalent  to  equation  (16). 

^    ,  c  1       ^  9"^       I  9^        I  9'2       I  9^  , 

Prob.  105.  Solve  — r-^ r:T-  =  -^:ri 1^   ^y   transposi- 

X  dx      X  dx      y  9/      y  dy 

tion  to  the  independent  variables  x^  and  y . 

Prob.  .06.  Solve  ^'g  +  2.yg^^  +yg-  =  o. 


374  GRASSMann's  space  analysis.  [Chap.  VIII. 


Chapter    VIII. 
GRASSMANN'S   SPACE  ANALYSIS. 


By  Edward  W.  Hyde, 

Professor  of  Mathematics  in  the  University  of  Cincinnati. 


Art.  1.    Explanations  and  Definitions. 

The  algebra  with  which  the  student  is  already  familiar  deals 
directly  with  only  one  quality  of  the  various  geometric  and 
mechanical  entities,  such  as  lines,  forces,  etc.,  namely,  with 
their  magnitude.  Such  questions  as  How  much?  How  far? 
How  long?  are  answered  by  an  algebraic  operation  or  series  of 
operations.  Questions  of  direction  and  position  are  dealt  with 
indirectly  by  means  of  systems  of  coordinates  of  various  kinds. 
In  this  chapter  an  algebra*  will  be  developed  which  deals 
directly  with  the  three  qualities  of  geometric  and  mechanical 
quantities,  viz.,  magnitude,  position,  and  direction.  A  geomet- 
ric quantity  may  possess  one,  two,  or  all  three  of  these  prop- 
erties simultaneously ;  thus  a  straight  Hne  of  given  length  has 
all  three,  while  a  point  has  only  one. 

The  geometric  quantities  with  which  we  are  to  be  concerned 
are  the  point,  the  straight  line,  the  plane,  the  vector,  and  the 
plane-vector. 

When  the  word  "  line  "  is  used  by  itself,  a  "  straight  line  " 
will  be  always  intended.  A  portion  of  a  given  straight  line  of 
definite  length  will  be  called  a  "  sect  "  ;  though  when  the  length 

*  The  algebra  of  this  chapter  is  a  particular  case  of  the  very  general  and 
comprehensive  theory  developed  by  Hermann  Grassmann,  and  published  by 
him  in  1844  under  the  title  "Die  lineale  Ausdehnungslehre,  ein  neuer  Zweig 
der  Mathematik."     He  published  also  a  second  treatise  on  the  subject  in  1862. 


Art.  2.J  EXPLANATIONS    AND    DEFINITIONS.  375 

of  the  sect  is  a  matter  of  indifference,  the  word  line  will  fre- 
quently be  used  instead.  Similarly,  a  definite  area  of  a  given 
plane  will  be  called  a  "  plane-sect." 

If  a  point  recede  to  infinity,  it  has  no  longer  any  significance 
as  regards  position,  but  still  indicates  a  direction,  since  all  lines 
passing  through  finite  points,  and  also  through  this  point  at 
infinity,  are  parallel.  Similarly,  a  line  wholly  at  infinity  fixes 
a  plane  direction,  that  is,  all  planes  passing  through  finite 
points,  and  also  through  this  line  at  infinity,  are  parallel.  Thus 
a  point  and  line  at  infinity  are  respectively  equivalent  to  a  line 
direction  and  a  plane  direction. 

A  quantity,  possessing  magnitude  only  will  be  termed  a 
"scalar  "  quantity.  Such  are  the  ordinary  subjects  of  algebraic 
analysis,  ^,  x,  sin  6^  log^,  etc.,  and  they  may  evidently  be  in- 
trinsically either  positive  or  negative. 

The  letter  T  prefixed  to  a  letter  denoting  some  geometric 
quantity  will  be  used  to  designate  its  absolute  or  numerical 
magnitude,  always  positive.  Thus,  if  Z  be  a  sect,  and  /^a  plane- 
sect,  then  TL  is  the  length  of  Z,  and  TP  is  the  area  of  P.  That 
portion  of  a  geometric  quantity  whose  magnitude  is  unity  will 
be  called  its  *'  unit,"  and  will  be  indicated  by  prefixing  the 
letter  U\  thus  UL  =  unit  o(  L  =  sect  one  unit  long  on  line  Z.* 
Hence  we  have  TL  .  UL  =  L. 

Art.  2.    Sum  and  Difference  of  Two  Points. 

In  geometric  addition  and  subtraction  we  shall  use  the  or. 
dinary  symbols  -|-,  — ,  =,  but  with  modified  significance,  as  will 
appear  in  the  development  of  the  subject. 

Every  mathematical,  or  other,  theory  rests  on  certain  fun- 
damental assumptions,  the  justification  for  these  assumptions 

*  The  word  "scalar  "and  the  use  of  the  letters  7"  and  (7,  as  above,  were 
introduced  by  Hamilton  in  his  Quaternions,  T^stands  for  tensor,  i.e.,  stretcher, 
and  TL  is  the  factor  that  stretches  L/L  into  L.  The  notation  |  L  |  for  absolute 
magnitude  is  not  used,  because  the  sign  |  has  been  appropriated  by  Grassmann 
to  another  use. 


376  grassmann's  space  analysis.  [Chap.  VIII. 

lying   in   the   harmony  and    reasonableness   of   the    resulting 
theory,  and  its  accordance  with  the  ascertained  facts  of  nature. 

Our  first  assumption,  then,  will  be  that  the  associative  and 
commutative  laws  hold  for  geometric  addition  and  subtrac- 
tion, that  is,  whatever  A,  B,  C  may  represent,  we  have 

A+B  +  C={A-{-B)-{-C=A  +  {B-{-C) 

=  A  +  C+B  =  {AJrC)  +  B. 

We  shall  also  assume  that  we  always  have  A  —  A  =  o,  and 
that  the  same  quantity  may  be  added  to  or  subtracted  from 
both  sides  of  an  equation  without  affecting  the  equality. 

Now  let/»j,/2  be  two  points,  and  consider  the  equation 

A  +A  -  A  =A  +  (A  -A)  -A-  (O 

In  this  form  we  have  an  identity.     Write  it,  however,  in  the 
form 

A  -A  +  A  =  (A  -  A)  +  A  =  A  >  (2) 

and  it  appears  that/,  —  p,  is  an  operator  that  changes/,  into 
p^hy  being  added  to  it.  Conceive  this  change  of/,  into/,  to 
take  place  along  the  straight  line  through  /,  and  />, ;  then  the 
operation  is  that  of  moving  a  point  through  a  definite  length 
or  distance  in  a  definite  direction,  namely,  from/,  to/,.  This 
operator  has  been  called  by  Hamilton  ''  a  vector,"  *  that  is,  a 
carrier,  because  it  carries/,  rectilinearly  to  /,.  Grassmann  gives 
to  it  the  name  Strecke,  and  some  writers  now  use  the  word 
"  stroke  "  in  the  same  sense. 

Again,/,—/,  is  the  difference  of  two  points,  and  the  only 
difference  that  can  exist  between  them  is  that  of  position,  i.e. 
a  certain  distance  in  a  certain  direction. 

Hence  we  may  regard/,  — /,  as  a  directed  length,  and  also 
as  the  operator  which  moves  /,  over  this  length  in  this  direc- 
tion.    Writing/,  — /,  =  e,  equation  (2)  becomes 

A  +  e=A.  •      (3) 

*  See  the  first  of  Hamilton's  Lectures  on  Quaternions,  where  a  very  full 
discussion  of  equation  (2)  will  be  found.     Also  Grassmann  (1862),  Art.  227. 


Art.  3.]  SUM    AND    DIFFERENCE    OF    TWO    POINTS.  377 

Thus  the  sum  of  a  point  and  a  vector  is  a  point  distant  from 
the  first  by  the  length  of  the  vector  and  in  its  direction. 

Since  p^—p^—  —  (/,  —  A)»  it  appears  that  the  negative 
of  a  vector  is  a  vector  of  the  same  length  in  the  opposite 
direction. 

If  /a  — /i  =  o,  or  /,  =A>  A  iTiust  coincide  with  A  because 
there  is  now  no  difference  between  the  two  points. 

The  question  arises  as  to  what,  if  any,  effect  the  operator 
A— A  should  have  on  any  other  point A>  that  is,  what  is  the 
value  of  the  expression  A  —pi  +A  ^ 

We  will  assume  that  it  is  some  point  A*  so  that  we  have 
A— AH-A=A> 
or  A-A=A-A-  (4) 

This  implies  that  the  transference  from  A  to  A  is  the  same 
in  amount  and  direction  as  that  from  A 
to  A'  that  is,  that  A>  A' A' A  ^re  the 
four  corners  of  a  parallelogram  taken  in 
order.  Thus  equal  vectors  have  the  same  /^ 
length  and  direction,  and,  conversely, 
vectors  having  the  same  length  and  direction  are  equal. 

Note  that  parallel  vectors  of  equal  length  are  not  neces- 
sarily equal,  for  their  directions  may  be  opposite. 

Equation  (4)  may  also  be  written 

A+A=A+A,  (5) 

so  that,  whatever  meaning  may  be  assigned  to  the  sum  of  two 
points,  if  we  are  to  be  consistent  with  assumptions  already 
made,  we  must  have  the  sum  of  either  pair  of  opposite  corner- 
points  of  a  parallelogram  equal  to  the  sum  of  the  other  pair. 
The  sum  cannot  therefore  depend  on  the  actual  distances 
apart  of  the  points  forming  the  pairs,  for  the  ratio  of  these  two 
distances  may  be  made  as  large  or  as  small  as  we  please. 

If  «  be  a  scalar  quantity,  ne  will  denote  that  the  operation 
€  is  to  be  performed  n  times  on  a  point  to  which  ne  is  added, 
that  is,  the  point  will  be  moved  n  times  the  length  of  e;  hence 


378  grassmann's  space  analysis.  [Chap.  VIIL 

ne  is  a  vector  n  times  as  long  as  e,  and  having  the  same  or  the 
opposite  direction  according  to  the  sign  of  n. 

In  the  figure  above,  let 

A-A  =  ei,   A-/i  =  e„    A-A=e3,   A— A  =  e4. 
Then 

e,  +  e,  =A  -P.+P.  -A  =A  -A  +A  -A  =P-P.  =  ^3,  (5) 
since,  by  Art.  4,^3  —p,=p,  —  A- 

Also,  e,  -  e,  ^A  —  A  =  e,.  (6) 

Hence,  if  two  vectors  are  drawn  outwards  from  a  point,  and 
the  parallelogram  of  which  these  are  two  adjacent  sides  is  com- 
pleted, then  the  two  diagonals  of  this  parallelogram  will  repre- 
sent respectively  the  sum  and  difference  of  the  two  vectors^ 
the  sum  being  that  diagonal  which  passes  through  the  origin 
of  the  two  vectors,  and  the  difference  that  which  passes  through 
their  extremities.* 

Again,  A  - p,  +/,  -/,  +A  "A  =  O  =  e,  +  e,  +  (-  ej  ; 
hence  the  sum  of  three  vectors  represented  by  the  sides  of  a 
triangle  taken  around  in  order  is  zero. 

Similarly,  if /,,/'2,  . .  ./«  be  any  n  points  whatever  taken  as 
corners  of  a  closed  polygon,  we  shall  have 
(A-A)+(A-A)+(A-A)+---+(A-A-x)+(A-A)=o; 
that  is,  the  sum  of  vectors  represented  by  the  sides  taken  in 
order  about  the  polygon  is  zero.  By  "taken  in  order"  is  not 
meant  that  any  particular  order  of  the  points  must  be  observed 
in  forming  the  polygon,  which  is  evidently  unnecessary,  but 
simply  that,  when  the  polygon  is  formed,  the  vectors  will  be 
the  operators  that  will  move  a  point  from  the  starting  position 
along  the  successive  sides  back  to  this  position  again,  so  that 
the  final  distance  from  the  starting-point  will  be  nothing. 


Art.  3.    Sum  of  Two  Weighted  Points.! 

Consider  the  sum  ^lA  +  ^aA^  i"  which  ;;^,  and  m^  arescalars, 
that  is,  numbers,  positive  or  negative,  and  /,,  /,  are  points. 

*  Grassmann  (1844),  §  15. 

f  Grassmann  (1844),  §  95,  and  (1862),  Art.  227. 


Art.  3.]  SUM  OF  two  weighted  points.  379 

The  scalars  m^  and  m^  will  be  regarded  as  values  or  weights 
assigned  to  the  points /^  and/,.  When  any  weight  is  of  unit 
value  the  figure  I  will  be  omitted,  so  that  p  means  i/,  and  is 
called  a  unit  point.  Occasionally,  however,  a  letter  may  be 
used  to  denote  a  point  whose  weight  is  not  unity. 

To  assist  his  thinking,  the  reader  may  consider  the  weights 
initially  as  like  or  unlike  parallel  forces  acting  at  the  points. 

In  order  to  arrive  at  a  meaning  for  the  above  expression 
we  shall  make  two  reasonable  assumptions,  which  will  prove  to 
be  consistent  with  those  already  made,  viz.,  first,  that  the  sum  is 
a  point,  and  second,  that  its  weight  is  the  sum  of  the  weights 
of  the  two  given  points.  Denoting  this  sum-point  by/,  we 
write 

^.P.  +  ni^P^  =  (m,  +  m^p.  (7) 

Transposing, we  have  mj^p^  —p)  =  mj^p  —/a),  or 

Both  members  of  (8)  are  vectors,  and,  being  equal,  they  must, 
by  Art.  4,  be  parallel.  This  requires  that  /  shall  be  collinear 
with  /,  and  p^.  Also,  since  p^  —  p  and  p  —  p^  are  vectors  whose 
lengths  are  respectively  the  distances  irova p^.to p  and  from/ 
to  /„  it  follows  that  these  distances  are  in  the  ratio  of  m^  to  m^. 
Hence,  /  is  a  point  on  the  line  /j/,  whose  distances  from/, 
and  /,  are  inversely  proportional  to  the  weights  of  these  points. 
We  shall  call  /  the  mean  point  of  the  two  weighted  points. 
If  m^  and  m^  are  both  positive,  (8)  shows  that/  must  lie  be- 
tween /j  and/, ;  but  if  one,  say  m^y  is  negative,  let  m^-=i—m^. 
Thus 

^X/i  -?)  =  ^/(  A  -  Pi  (9> 

and  /  is  on  the  same  side  of  each  point,  that  is,  its  direction 
from  each  point  is  the  same.  Also,  since  its  distances  from  the 
two  points  are  inversely  as  their  weights,  /  must  be  nearest 
the  point  whose  weight  is  greatest. 


380  grassmann's  space  analysis.  [Chap.  VIII. 

Case  when  m^-{-m^  =  o,  or  m^  =  —  m  .^ — With  this  con- 
dition equations  (7)  and  (8)  become 

^lA  +  ^.A  =  ^i(A  -  A)  =  o  -A  (10) 

and  p-p^^-p—p^,  (11) 

Thus/  is  in  the  same  direction  from  each  point,  that  is,  not 
between  them,  and  yet  is  equidistant  from  them.  This  re- 
quires either  that  the  two  points  shall  coincide,  that  is,/,,  =/,, 
which  evidently  satisfies  (10)  and  (11);  or  else,/,  and/,  being 
"different  points,  that/  shall  be  at  an  infinite  distance.  Thus 
the  sum  is  in  this  case  a  point  of  zero  weight  at  infinity.f 
Eq.  (10)  shows  that  a  zero  point  at  infinity  is  equivalent  to  a 
vector,  or  directed  quantity,  as  stated  in  Art.  i.  It  has  been 
rshown  in  Art.  2  that  /,  =/,  is  the  condition  that  /,  and  /, 
coincide ;  let  us  consider  the  equality  of  weighted  points  in 
general,  say  ^lA^^aA.  Hence,  by  (7),  there  is  found 
^^1 A  ~  ^sA  =  (^1  ~  ^2)/  =  o;  hence,  since  /  cannot  be  zero, 
m^—  m^=  o,  or  m^  =  m^\  and  therefore  w^A  ~~  A)  =  o,  or, 
since  m^'%Xi^  /,  — /,  =0,  that  is,  /,  •=^p^.  Therefore,  if  any 
two  points  are  equal,  their  weights  must  be  the  same  and  their 
positions  identical,  that  is,  they  are  the  same  point. 

Exercise  i.-^— To  find  the  sum  and  difference  of  the  two 
weighted  points  3/,  and  /, : 

3/1  +  A  =  4A         3A  -  A  =  2/', 

and  the  mean  points  are  as  shown  in 
3_         1  3    -  ^ 

r  *p      ;  I  the   figure.     The  reciprocals  of  the 

^^'  3^'    *^  ^'distances  of  /,  /„  and/  from  /,, 

viz.,  \,  J-,  ^,  are  in  arithmetical  progression,  hence  the  points 
form  a  harmonic  range. 

Exercise  2. — Given  a  circular  disk  with  a  circular  disk  of 

*Grassmann  (1862),  Art.  222. 

t  Compare  the  case  of  the  resultant    of    unlike  parallel  forces  of  equal 
magnitude. 


Art.  4.]  SUM  OF  two  weighted  points.  381 

half  its  radius  removed,  as  in  the  figure ;  to  find  the  centroid. 
of  the  remaining  portion.  — 

Take  p^  at  center  of  large  circle,  p^  at  center 
of  small  circle,  and  /,  at  the  point  of  contact ;  ^^M  p    \p^ 
then /j  =  J( />,  +  A)-     The  areas  of  the  two  cir- 
cles are  as  i  14;  call  them  i  and  4.     Then  it  is  as 
if  there  were  a  weight  4  at  /„  and  a  weight  —  i  at  /j ;  hence 
~P  =  [4P.  -  i{P^  +  A)]  -  3  =  {7P^  -  /J  -  6. 

Prob.  I.  Show  that  p^tp^,  tn^p^  -\-  m^p^y  and  tn^p^  —  /«,/,  are 
four  points  forming  a  harmonic  range. 

Prob.  2.  An  inscribed  right-angled  triangle  is  cut  from  a  circular 
disk  ;  show  that  the  centroid  of  the  remainder  of  the  disk  is  at  the 
point 

(t^u  —  2  sin  2a)  p^  —  p^  sin  2a 
2i(jt  —  sin  2a)  * 

if /j  is  the  center  of  the  circle,  p^  the  opposite  vertex  of  the  triangle, 
and  OL  one  of  its  angles. 


Art.  4.    Sum  of  any  Number  of  Points. 

As  in  the  last  article  we  assume  the  sum  to  be  a  point 
whose  weight  is  equal  to  the  sum  of  the  weights  of  the  given 
points ;  thus, 

2mp=p2m,  (12) 

H 

Let  e  be  some  fixed  point,   and  subtract  e^m  from  both 

1 
sides  of  (12) ;  thus  we  have 

^m{p  -e)={p-  e)2m,  (13) 

1  1 

an  equation  which  gives  a  simple  construction  for/. 

n  n 

If  2m  =  o,  then  m^  =  —  2m,   and 
•  1  8 

f           ^"^^   ^ 
2mp  =  m^p^  -\-  2mp  =  m\   A ;, —    ,  (14) 

'  '  \  2m 


382  grassmann's  space  analysis.  [Chap.  VIII. 

so  that  the  sum  becomes  the  difference  of  two  unit  points,  or 
a  vector  whose  direction  is  parallel  to  the  line  joining  p^  with 
the  mean  of  all  the  other  points  of  the  system,  and  whose 
length  is  m^  times  the  distance  between  these  points.  Since 
any  point  of  the  system  may  be  designated  as  p^,  it  follows 
that  the  line  joining  any  point  of  the  system  to  the  mean  of 

all  the  others  is  parallel  to  any  other  such  line.     If  ^mp  =  o, 

equation  (14)  shows  that/,  is  the  mean  of  all  the  other  points 
of  the  system,  and,  since  any  one  of  the  points  may  be 
taken  as  />,,  any  point  of  the  system  is  the  mean  of  all  the 
others. 

Let  «  =  3  in  (12)  and  (13);  then 

^,  A  +  ^^^A  +  ^3/3  =  (?^^i  +  ^a  +  ^dpy  (l  5) 

^^i(A  -  ^)+  ^iP-^)  +  ^8( A-^)=(^i+^^.+^3)(/-^),    (16) 

andp  is  on  the  line  joining  the  point  m^p^  -f"  m^p^  with/3,  and 
therefore  inside  the  triangle  pip^p,  if  the  m's  are  all  positive. 
If  m^  be  negative  and  numerically  less  than  m^-\-m^j  then/ 
will  have  passed  across  the  line  p^p^  to  the  outside  of  the  tri- 
angle. If  m^  and  w^  are  negative  and  their  sum  numerically 
less  than  m^y  then  /  will  have  passed  outside  the  triangle 
through /a,  i.e.,  it  will  have  crossed /,/3  and  p^P^-  The  point 
€  must  evidently  always  be  in  the  plane /,/,/,. 

As  a  numerical  example  let  m^  =  3,  m^  =  4,  m^=  —  5,  so 
that  (16)  becomes 

p-e  =  HP,  -e)+  2(A  -e)-  f(A  -  e). 
Now,  since  e  may  be  any  point  whatever,  put  e=p^;    then 
p—ps  =  f(/i  —A)  +  ^{A  — A)'  ^"^  the  construction  is  shown 
in  the  figure.    p,-p^  =  f(/,  -p^),  and  p  - /,  =  2(/,  -p,). 

As  another  example  take/  =  4/,  +  5/,  —  2/3  —  6p^,  or,  by 
{13),  making  ^=  A, 

/  -- A  =  4(A  -A)  +  5(A  -  A)  -  2(A  -A) 
=  A-A+A-A+/-A- 


Art.  4.]  SUM  OF  any  number  of  points.  383 

When  any  number  of  geometric  quantities  can  be  connected 
with  each  other  by  an  equation  of  the  form  '2,mp  =  o,  in  which 
the  m's  are  finite  and  different  from  zero,  then  they  are  said  to 
be  mutually  dependent,  that  is,  any  one  can  be  expressed  in 
terms  of  the  others.     If  no  such  relation  can  exist  between  the 


..^5 


:^Ps 


/ 
/ 


P' 


•2P^     / 
•'Si?, 


quantities,  they  are  independent.     We  obtain  from  what  has 
preceded  the  following  conditions; 
That  two  points  shall  concide, 

^iA  +  ^,A  =  0.  (17) 

That  three  points  shall  be  collinear, 

^x/i  +  ^.A  +  ^^3  A  =  0.  (18) 

That  four  points  shall  be  coplanar, 

»^,/i  +  ^,A  +  ^sA  +  W4A  =  o-  (19) 

It  follows  that  three  non-collinear  points  cannot  be  con- 
nected by  an.  equation  like  (18)  unless  each  coefficient  is 
separately  zero.  Similarly  four  non-coplanar  points  cannot  be 
connected  by  an  equation  like  (19)  unless  each  coefficient  is 
separately  zero. 

The  significance  of  these  statements  will  be  presently  illus- 
trated. 

The  following  are  corresponding  equations  of  condition  for 
vectors : 

That  two  vectors  shall  be  parallel, 

n,e,  +  ?/,€,  =  o.  (20) 


384 


GRASSMANN  S   SPACE    ANALYSIS. 


[Chap.  VIII. 


That  three  vectors  shall  be  parallel  to  one  plane, 

n,e,-\-n^e^-\-n,e,  =  o.  (21) 

These  conditions  follow  from  the  results  of  Art.  2,  or  from 
equations  (17)  and  (18)  by  regarding  the  e's  as  points  at  infinity. 
If  in  addition  to  (21)  we  have 

^1  +  «.  +  «3  =  o,  (22) 

the  extremities  of  the  three  vectors,  if  radiating  from  a  point, 
will  be  collinear :  for,  let  ^^  ...  ^3  be  four  points  so  taken  that 
^,  —  ^0  =  ^1 ,  ^2  —  ^0  =  ^a »  ^3  —  ^0  =  ^3 '  then  (21)  becomes 

«i(^i  -  O  +  «a(^a  -  ^0)  +  ^3(^3  -  O  =  o, 

or  by  (22)  n,e,  +  n^e^  +  n,e,  =  o, 

which  by  (18)  requires  ^j,  e^,  e^  to  be  collinear. 

It  may  be  shown  similarly  that 

^«e  =:  ^«  :=  O 


(23) 

are  the  conditions  that  four  vectors  radiating  from  a  point  shall 
have  their  extremities  coplanar. 

Exercise  3. — Given  a  triangle  e^e^e^  and  a  point  /  in  its 

plane;  pe^  cuts  e,e^  in  q,, 
pe^  cuts  e^e^  in  q^ ,  pe^  cuts 
^0^1  in  ^,,^,^,  cuts  ^/,  in  A, 
q^q,  cuts  e^e,  in  p, ,  and  q,q, 
cuts  e^e^  in  /, :  to  show  that 
p^,  p^,  and  /,  are  collinear. 

"Lttp^n.e^+n.e.+n^e^; 
ps^     then  q,,q,,  q^  coincide  re- 
spectively with  n^e^  +  n^e^ , 
n^e^  -\-  n^e^ ,  and  n^e^  -\-  n^e^  because  p  lies  on  the  line  joining  e^ 
with  q^ ,  etc.     Hence,  if  x^,  ^, ,  jVo >  jVi  are  scalars, 

A  =  ^0^0  +  ^1^,  =  yW.  +  «»^.)  +  ji(«a^a  +  «o^o) ; 

hence      {x^  -  y,n,)e,  +  {x,  —  y,n,)e,  —  njj^  +yy^  =  o. 

Now  the  ^'s  are  not  collinear,  and  yet  are  connected  by  a 


Art.  4.]  SUM    OF    ANY    NUMBER    OF    POINTS.  385 

relation    of   the   form  of   equation  (i8);  hence,  as  was  there 
shown,  each  coefficient  must  be  zero ;  accordingly 

•^0  -y^n,  =  -^i-jj^o^i  =  Jo  +^1  =  o, 

whence  we  find  x^:x^  =  n^:  —  n^. 

hence  («„  —  «,)/),  =  n^e^  —  n^e^ ,  and  similarly 

{n-n^p,  =  n,e,  -  n,e,,     (n-n^p^  =  n^e^-n,e,. 

Adding,  we  have 

(«i  -  «,)A  +  («,  -  «o)/i  +  (^0  -  «i)A  =  o ; 
therefore,  by  (i8),  /o » /i  >  A  ^^^  collinear. 


Exercise  4. — Let  /  =  '^ne  H-  ^n  be  any  point  in  the  plane 

of  the  triangle  e^e^e^ :  show  that  lines  through  the  middle 
points  of  the  sides  e^e^ ,  ^/„ ,  and  e^e^  of  the  triangle  parallel 

to  e^p,  e^p,  and  e^p  meet  in  a  point 

2 

/  =  [(«!  +  «>o  +  («» +  ^oV,  +  («o  +  «,K]  -^  2:s«. 

By  the  conditions  the  vector  from  the  middle  point  of  e^e^ 
to/'  is  a  multiple  of  the  vector  e^—  p\  hence 

/-i(^. +  0  =  ^(^0-/)     or 

/  =  K^:  +  ^0  +  ^{e,  -p)  =  i{e,  +  e,)  -\-y{e,  -  p\ 

or,  substituting  value  of/, 

/  =  \{e,-\-e::)■^rx{e-:2ne-r^n)  =  i{e,+ e,)+y{e,--.2ne-i-2n). 

hence         [{x  —  i)2n  4-  n^y  —  x)]e,  +  n,(j;  —  x)e, 

+  l{i  -  y)^n  +  nly  -  x)-]e,  =  o ; 

therefore,  as  in  the  previous  exercise,  each  coefficient  must  be 
zero,  whence  x  =  j/  =  i,  and  substituting  we  find  p'  as  above. 
It  follows  also  that  the  distances  of  p'  from  the  middle  points 
of  the  sides  are  the  halves  of  the  distances  of/  from  the  oppo- 
site vertices. 


Prob.  3.  Show  that  e  =  ^'2e  is  collinear  with  p  and  /'  of  Exer- 


386 


grassmann's  space  analysis. 


[Chap.  VIII. 


cise  4.  Also  that,  by  properly  choosing/,  it  follows  that  e  is  col- 
linear  with  the  common  point  of  the  perpendiculars  from  the  vertices 
on  the  opposite  sides,  and  the  common  point  of  the  perpendiculars 
to  the  sides  at  their  middle  points. 

Prob.  4.  Given  two  circles  and  an  ellipse,  as  in  the  figure,  with 
centers  at  ^0  j  A  j  ^^^  Px-  Radii  of  circles  4  and 
I,  axes  of  ellipse  2  and  4,  small  circle  and  ellipse 
touching  large  circle  at  e^  and  e^  respectively, 
e^e^e^  an  equilateral  triangle:  show  that  the  cen- 
troid  of  the  remainder  of  the  large  circle,  after 
the  small  areas  are  removed,  will  be  at 

/  =  -i^^^^e,  -  A  -  2A)=tV(59^o  -  4^1  -  3^a)- 

Prob.  5.  If  a  sheet  of  tin  in  the  shape 
of  an  isosceles  triangle  be  folded  over  as  in 
the  figure,  show  that  its  centroid  is  given  by 

3^=  A-[35(^  +  ^,)  +  ii^J- 

Prob.  6.  If  a  tetrahedron  e^e^e^e^  have  a 
tetrahedron  of  \  of  its  volume  cut  off  by  a 
plane  parallel  to  e^e^e^ ,  and  one  of  -^  of  its 
volume  cut  off  by  a  plane  parallel  to  e^e^e^ , 
show  that  the  centroid  of  the  remaining  solid  is  at 

/=  ^k(227^o  +  iTS'^s  +  239(^1  +  O  )• 


R 


Art.  5.    Reference  Systems. 


Let  p  be  any  unit  point,  e^,  ^, ,  e^  three  fixed  unit  points, 
and  w,  x^  y  scalars ;  then,  v^rriting 

p  =  we,-\-  xe^  +  ye^ ,  (24) 

we  must  have  also,  because  /  is  a  unit  point, 

w-\-x^y=\,  (25) 

and  p  is  the  mean  of  the  freighted  points  we^ ,  xe^ ,  ye^.  The 
point  /  may  occupy  any  position  whatever  in  the  plane  e^e^e^ ; 
for  it  is  on  the  line  joining  we^  +  xe^  with  e^,  and  by  varying 

ID 

y  and  w-\-  x,  —  remaining  constant,  /  may  be  moved  along 

"^  X 


Art.  5.]  reference  systems.  387 

this  line  from  —  oo  to  -|-  °^  5  while  by  varying  the  ratio  —  the 

point  we^  + -^^i  ^^'^V  be  moved  from  —  oo  to  +  oo  along  e^e^y 
and  thus  the  first  line  will  be  rotated  through  i8o  degrees,  and 
/  may  thus  be  given  any  position  whatever  in  the  plane. 

A  system  of  unit  points  to  which  the  positions  of  other 
points  may  be  referred  is  called  a  reference  system,  and  the 
triangle  e^e/^  is  a  reference  triangle.  For  reasons  that  will  ap. 
pear  later,  the  double  area  of  this  triangle  will  be  taken  as  the 
unit  of  measurement  of  area  for  a  point  system  in  two-dimen- 
sional space. 

Similarly,  in  solid  space,  taking  a  fourth  point  ^3,  we  write 
p  —we,-{-  xe,  -\-ye^  +  ze^,  (26) 

which  implies  also      w-{-x-\-y-\-z-=ii\  (27) 

and  p  may  be  shown  as  above  to  be  capable  of  occupying  any 
position  whatever  in  space  by  properly  assigning  the  values  of 
w,  X,  y,  z\  so  that  ^„  .  .  .  e^  form  a  reference  system  for  points 
in  three-dimensional  space.  The  tetrahedron  e^e^e^e^  is  called 
the  reference  tetrahedron,  and  six  times  its  volume  will  be 
taken  as  the  unit  of  volume  for  a  point  system  in  three-dimen- 
sional space. 

Eliminating  w  between  (24)  and  (25),  we  have 

/  =  ^0  +  Ae.  -  ^0)  +A^.  -  ^o\  (28) 

from  which  it  may  also  be  easily  seen  that  /  may  be  any  point 
in  the  plane  e,e,e,.  Writing/  -  e^  =  p,  e,  -  e,  =  e,,  e,  — e,  =  e„ 
(28)  becomes  p  =  ;ire,  +  ye, ,  (29) 

and  e,,  e,  form  a  plane  reference  system  for  vectors. 
Similarly,  from  (26)  and  (27)  we  find 

p  =  X6,+ye,-i-Z€„  (30) 

and  e,,  €,,  e,  are  a  reference  system  for  vectors  in  solid  space, 
any  vector  whatever  being  expressible  in  terms  of  these 
three. 

If,  in  equations  (25)  and  (26),  the  reference  vectors  are  of 


388 


GRASSMANN  S    SPACE    ANALYSIS. 


[Chap.  VIII. 


unit  length  and  mutually  perpendicular,  we  have  unit,  normal 
reference  systems,  and  in  this  case  i. ,  i,,  i^  will  generally  be  used 
instead  of  e^  e,,  eg. 

Exercise  5. — To  change  from  one  reference  system  to  an- 
other, say  from  e^,  e^,  e^  to  ^/,  ^/,  e^. 

The  new  reference  points  must  be  connected  with  the  old 
ones  by  equations  such  as 

Then  any  point  /  =  x^e^  -\-  x^e^  +  -^a^s  ^^11  be  expressed  in 
terms  of  the  new  reference  points  by  substituting  the  values  of 
^0,  etc.,  as  given.  If  ^/,  ^/,  e^  are  given  in  terms  of  the  old 
points,  e^,  e^,  e^  may  be  found  by  elimination.  Thus,  if  e^'=2ley 
e^  =  ^me^  e^  =  'Ene^  we  have  at  once 


h 

A 

/, 

e: 

I, 

h 

m. 

m. 

m^ 

^0  = 

e; 

m^ 

m. 

'^K 

«i 

«, 

e: 

^1 

n. 

with  similar  values  for  e^  and  e^. 

As  a  numerical  example  let  the  new  reference  triangle  be 
formed  by  joining  the  middle  points  of  the  sides  of  the  old  one. 
Then   ej  =  i(e,  +  e,\  e/  =  i(e,  +  e,),  e^  =  i{e,  +  ^J  ;  whence 

^0  =  -  ^/  +  ^/  +  ^/,    ^i  =  ^0'  -  ^/  +  ^/>    ^a  =  ^0'  +  ^/  —  ^/. 
Thus  /  =  x,e^  +  ;ir/j  +  ;i;,^, 

=  (-  ^0  +  ^x  +  ^a)^o'  +  (^0  -  ^i  +  ^,W  +  (^0  +  ^1  —  •^,)^/- 

Exercise  6. — Three  points  being  given  in  terms  of  the  refer- 
ence points  ^,,  e,,  e^,  find  the  condition  that  must  hold  between 
their  weights  when  they  are  collinear. 


Let  p,  =  2/e,  /,  =  ^me,  p,  =  2ne;  then,  k,,  ^„  k^  being 

000 

scalars,  we  must  have  for  coUinearity,  by  (18), 


Art.  5.]  REFERENCE    SYSTEMS.  389 

that  is,  k^'2le  +  k'2^me  +  k^ne  =  o, 

whence         {k,l,  +  k^m,  +  Kn^e,  +  {kj^  +  k^m^  +  >^,«,)^, 

+  {KK  +  >^/^^.  +  K^^^^  —  o, 

and,  as  e^,  e^,  e^  are  not  coUinear,  the  coefficients  must  be  zero, 
by  Art.  4 ;  hence 

and,  by  ehmination  of  the  ^'s, 

/o       Wo       ^0 


O, 


(31) 


which  is  the  required  condition  of  collinearity. 


Prob.  7.  If  /  =  3^0  -  e,  —  e^,  4^/  =  3^,  +  e^ ,  4^/  =  3^,  +  ^^ , 


4^/  =  3^  +  ^1  >  show  that  ^p 


19^0'  -  3^/  +  29*?/. 


Prob.  8.  Find  the  condition  that  four  points  ^ke^  '2le^  ^me,  2ne 

0000 

shall  be  coplanar.     Ans.  [k^ ,  A  >  ^^  >  ^sl  ==  ^• 

Prob.  9.  U  J>  =  we^ -{- xe^ -{-ye^f  and  there  exist  between  the 
scalars  w,  x^y  a  linear  relation  such  as  Aw  +  ^^  +  O'  =  o>  ^>  ^> 
C being  scalar  constants,  show  that/ will  always  lie  on  a  straight 
line  which  cuts  the  reference  lines  in  Ae^  —  Be^^ ,  Ae^  —  Ce^ ,  and 
Ce^  —  Be^.  Consider  the  special  cases  when  A  ^=  B^  B  =  C,  C=  Ay 
A  =  B  =  C,  A  =z  o^  B  =  Of  and  C  =  o. 

Prob.  10.  If  /  =  we^  -{-  xe^  -\-ye^  +  -s^^s  >  and  there  exist  also  an 
equation  Aw  +  Bx  +  Q'  +  B>z  =  o,  show  that  /  will  lie  on  a  plane 

which    cuts   the   edges   of   the   reference  tetrahedron   in  -^^ j", 

6  € 

—  —  -j-f  etc.      Also,  if   a  second  relation  between  the   variables, 

such    as   A'w  +  B^x  -\-  C'y  -{-  D'z  =  o,  be   given,  then  /  Hes  on  a 
line  which  pierces  the  faces  of  the  reference  tetrahedron  in 


etc. 


^0 

^, 

^, 

^3 

^0 

^i 

A 

B 

C 

> 

D 

A 

B 

A' 

B' 

a 

D' 

A' 

B' 

390  GRASSM Ann's  space  analysis.  [Chap.  VIIU 

Art.  6.     Nature  of  Geometric  Multiplication.-^ 
The  fundamental  idea  of  geometric  multiplication  is,  that  a 

product  of  two  or  more  factors  is  that  which  is  determined  by 

those  factors. 

Thus,  two  points  determine  a  line  passing  through  them, 

and  also  a  length,  viz.,  the  shortest  distance  between  them ; 

hence /j/>,  =  L  is  the  sectf  drawn  from  p^  to/,,  or  generated 

by  a  point  moving  rectihnearly  from  /j  to  /,. 

The  student  should  note  carefully  the  difference  between 
p^p,  and/,  — /j ;  they  have  the  same  length  and  direction,  but 
the  sect/1/2  is  confined  to  the  line  through  these  two  points, 
while  the  vector/,  —  p^  is  not.  The  sect  has  position  in  addi- 
tion to  the  direction  and  length  possessed  by  the  vector. 

Again,  in  plane  space,  two  sects  determine  a  point,  the 
intersection  of  the  lines  in  which  they  lie,  and  also  an  area,  as 
will  appear  later,  so  that  L,L^  =  /,  in  which  /  is  not  in  general 
a  unit  point.  In  solid  space,  however,  two  lines  do  not,  in 
general,  meet,  and  hence  cannot  fix  a  point ;  but  two  sects,  in 
this  case,  determine  a  tetrahedron  of  which  they  are  opposite 
edges. 

It  appears,  therefore,  that  a  product  may  have  different 
interpretations  in  spaces  of  different  dimensions.  Hence  we 
will  consider  separately  products  in  plane  space,  or  planimetric 
products,  and  those  in  solid  space,  or  stereometric  products. 

Products  of  the  kind  here  considered  are  termed  "  com- 
binatory," because  two  or  more  factors  combine  to  form  a 
new  quantity  different  from  any  one  of  them.  This  is  the 
fundamental  difference  between  this  algebra  and  the  linear 
associative  algebras  of  Peirce,  of  which  quaternions  are  a 
special  case. 

Before  discussing  in  detail  the  various  products  that  may 
arise,  we  will  give  a  table  which  will  serve  as  a  sort  of  bird's-eye 
view  of  the  subject. 

*  Grassmann  (1844),  Chap.  2  ;  (1862),  Chap.  2. 
f  See  Art.  i. 


Art.  6.] 


NATURE    OF    GEOMETRIC    MULTIPLICATION. 


39^1 


In  this  table  and  generally  throughout  the  chapter  we  shall 
use/j/j,/,,  etc.,  for  points;  e,  e^,  e,,  etc.,  for  vectors ;  Z,  Z^, 
etc.,  for  sects,  or  lines;  rj,  tf^j  etc,,  for  plane-vectors  ;  and  P,  P^, 
etc.,  for  plane-sects,  or  planes.  Also/, /j,  etc.,  as  used  in  this 
table  will  not  generally  be  unit  points. 

The  products  are  arranged  in  two  columns,  so  as  to  bring 
out  the  geometric  principle  of  duality. 

Planimetric  Products. 


PJ.  =  L. 

L.K=P- 

PiP-iPz  =  area  (scalar). 

L^L^L^  =z  (area)''(scalar). 

pL  =  area  (scalar). 

Lp  =  area  (scalar). 

/.  •  ^.4  =  L. 

A  •/,/,  = /• 

p.p. -p.p.  =p- 

L,L, .  L^L,  =  L. 

p.p.  ■p.p. .  AA=(area)'(scalar). 

Z,Z,.Z,Z..^,Z.=^(area)'(scalar> 

e^e,  =  area  (scalar). 


Stereometric  Products. 


P.P.  =  L. 

P.P.  =  L. 

P.P.P.  =  P- 

P.P.P.^P- 

/lAAA  =  volume  (scalar). 

P,P,P,P,  =  (volume)'  (scalar)^ 

pP  z=  volume  (scalar). 

Pp  =  volume  (scalar). 

ZjZ,  =  volume  (scalar). 

ZjZ,  =  volume  (scalar). 

pL  =  Lp  =  P, 

PL  =  LP  =  p. 

P.PA  =  P' 

P'P.P.=P' 

p.P,P,P,=L. 

P'P.P.P3    =    ^' 

^'P;P,P.=P* 

L  .  P,P,P,  =  P. 

€,€,  =  1^. 

V.%  =  e. 

€,€,€3  =  volume  (scalar). 

T^^r/^r^^  =  (volume)'  (scalar). 

e.e,.e.e,  =  e. 

V.V.'V.V.  =  V' 

392  GRASSMAnn's  space  analysis.  [Chap.  VIII. 

Laws  of  Combinatory  Multiplication.  —  All  combinatory 
products  are  assumed  to  be  subject  to  the  distributive  law  ex- 
pressed by  the  equation 

A{B+C)=AB  +  AC. 

The  planimetric  product  of  three  points  or  of  three  lines, 
and  the  stereometric  product  of  three  points  or  planes,  or  of 
four  points  or  planes,  are  subject  to  the  associative  law.  That  is, 

In  Plane  Space  : 

AAA  =  AA  -A  =  A  •  AA ;    AAA  =  A  A  •  A  =  A  •  AA- 

In  Solid  Space  : 
/.AA  =  /.  -A/.  =  />,A  A ;    P.P,P,  =  P,  ■  P,P,  =  P,P,  ■  P.- 

AAAA=A-AAA=AA-AA; 

P,P^,P,  -  P. .  P,P,P,  =  PA .  P,P, 

The  commutative  law  of  scalar  algebra  does  not,  in  general, 
hold.  Instead  of  this,  in  the  products  just  given  as  being  asso- 
ciative, a  law  prevails  which  may  be  expressed  by  the  equation 

AB=-  BA, 

from  which  it  follows  that  the  interchange  of  any  two  single 
factors  of  those  products  changes  the  sign  of  the  product.* 

Since  vectors  are  equivalent  to  points  at  oo ,  the  associative 
law  holds  for  e^e^ej  and  rf^tf^rf^. 

Art.  7.    Planimetric  Products. 

Product  of  Two  Points.f— This  has  been  fully  defined  in 
Art.  6,  and  it  is  evident  from  its  nature  as  there  given  that 

AA=-AA.  (32) 

If  p^  =/„  this  becomes  p^p^  —  o,  which  must  evidently  be 
true,  since  the  sect  is  now  of  no  length. 

Also,  A(A  -A)  =AA  -AA  =  AA-  (33) 

*  Grassmann  (1862),  Chap.  3.         f  Grassmann  (1862).  Arts.  245,  246,  247. 


Art.  7.]  planimetric  products.  393 

But /a  — /j  is  a  vector,  say,  e  ;  hence 

/x^  =  /:  A ;  (34) 

•or  the  product  of  a  point  and  a  vector  is  a  sect  having  the  di- 
rection and  magnitude  of  the  vector  ;  or,  again,  multiplying  a 
vector  by  a  point  fixes  its  position  by  making  it  pass  through 
the  point. 

To  find  under  what  conditions  //'  will  be  equal  to  /,/,. 
Take  any  other  point/,  in  the  plane  space  under  consideration, 
and  write  p  =  x,p,  +  x^^-\-x^p^,  p'  =y,p,  +y^p^  -\- y^P^y  with 
the  conditions  for  unit  points  ^x  —  '^y  =  0. 

Then      //=       •   ^      p,p,-{-     J     '    /^3+     J     '    AA- 


If  this  is  to  reduce  to/,/,,  we  must  have  the  third  condition 
-^aJ^s  —  ^3  Ja  =  -^sJ^i  ~  ^iJ^3  =  0»  which  requires  that  x^=y,  =  o, 
unless  the  coefficient  of  /,/,  is  to  vanish  also.  Thus  //^  must 
be  in  the  same  straight  line  with/,/,.  If,  moreover,  in  addition 
jir,jj/,  —  x^y^  =  I,  we  shall  have//'  —  p^p^  Hence  pp'  is  equal 
to p^p.^  when,  and  only  when,  the  four  points  are  collinear,  and 
J>'  is  distant  from/  by  the  same  amount  and  in  the  same  direc- 
tion that  /,  is  from  /,. 

Product  of  Three  Points. — By  Art.  6  the  product  is  what 
is  determined  by  the  three  points.  In  solid  space  they  would 
fix  a  plane,  but,  as  we  are  now  confined  to  plane  space,  this  is 
not  the  case.  The  points  evidently  fix  either  a  triangle  or  a 
parallelogram  of  twice  its  area,  and  the  product  /,AA  will  be 
taken  as  the  area  of  this,  or  an  equivalent,  parallelogram. 

This  area  is  taken  rather  than  that  of  the  triangle,  because 
it  is  what  is  generated  by/,/,  as  it  is  moved  parallel  to  its 
initial  position  till  it  passes  through /g. 

We  have  /,/,/,  =/,  .p^,  =  -A  -AA  =  - P^P^Pi^  so  that 
if  we  go  around  the  triangle  in  the  opposite  sense  the  sign  is 
•changed.  As  this  product  possesses  only  the  properties  of  mag- 
nitude and  sign  it  is  scalar. 

Write/  =  ^xp,p'  =  '^yp.p"  =  ^zp ;  then 
111 


394  grassmann's  space  analysis.  [Chap.  VIII. 

x^  x^  X, 

PP'P"  =  y.  y-^  y^P.P.P^'^  (35) 


x^ 

X, 

-^3 

y. 

y. 

^3 

^1 

^. 

^3 

that  is,  any  triple  point  product  in  plane  space  differs  from  any 
other  only  by  a  scalar  factor.* 

Finally,    AAA  =  A(A  -  A)(A  -  A)  =  A^e^  (36) 

if  e  =  A  —A  and  e'  ^ p,  — A- 

Product  of  Two  Vectors. — Using  the  values  of  e  and  e' 
just  given,  we  see  that  e  and  e'  determine  the  same  paral- 
lelogram that  A>  A'  a"d  A  <^o ;  hence  the  meaning  of 
the  product  is  the  same  in  all  respects  in  two-dimensional 
space. 

We  shall  have  ee'  =  —  e'e,  for 

ee'  =  (A  -  AXA  -a)  =  -  (A  -P^){P.-P.)  =  -  e'e; 

since  we  have  shown  that  inverting  the  order  changes  the  sign, 
in  a  product  of  points.  The  result  may  be  obtained  also  by 
regarding  e  and  e'  as  points  at  infinity,  or  by  consideration  of 
a  figure. 

As  we  have  seen  that  ee'  has,  in  plane  space,  precisely  the 
same  meaning  as  AAA  ^^  ^^y  write 
AAA=Aee'  =  €e' 

=  (A  -  /.)( A  -  P.)  =  P.P.  +  P^P.+ P.P.-  (37) 
Thus  the  sum  of  three  sects  which  form  the  sides  of  a  triangle, 
all  taken  in  the  same  sense  as  looked  at  from  outside  the 
triangle,  is  equal  to  the  area  of  the  triangle. 

Product    of   Two    Sects. — Any  two    sects    in  plane  space, 
Z,,  Z„  determine  a  point,  the  intersec- 
/     tion  of  the  lines  in  which  they  lie,  and 

^ /         an    area,  that  of  a  parallelogram  as  in 

"  '  '  the  figure.     Let  A  be  the  intersection, 

and  take  /,  and  A  so  that  L^  =  p^p^  and  Z,  =  /„ A*     The  area 

*  Grassmann  (1862),  Art.  255. 


Art.  7.]  planimetric  products.  395 

determined  by   L^  and  L^   is   then    the   same   that   we   have 
given  as  the  value  of /o/,/,.     We  write  therefore 

AA  =  AA  -AA  =  AAA .  A-  (38) 

The  third  member  of  (38)  is  not  to  be  regarded  as  derived 
from  the  second  by  ordinary  transposition  and  reassociation  of 
the  points,  for  the  associative  law  does  not  hold  for  the  four 
points  taken  together,  since /o/j/o ./,  =  o.  The  third  member 
simply  results  from  the  definition  of  L^L^f^  It  may  be  taken 
as  a  model  form  which  will  be  found  to  apply  to  several  other 
cases,  for  instance  to  (38)  when  points  and  lines  are  inter- 
changed throughout.    Thus,  if /j  =  L^L^  and  /,  =  L^L^  we  have 

/,  A  =  A^, .  L^L,  =  L^L^L^ .  Z,.  (39) 

For  take  //  and  //  so  that  A//  =  L^  and /,//=  L^ ;  p^p^  is 
evidently  some  multiple  of  Z„ ,  say  nL^ ;  hence 

AA  =  nL,  =  ^{PJ,  ^P.p:) .  (AA  .  AAO 
ft 

=  7.(AA//.A)-(AAA^A)>  by  (38), 

=  —a  •  AAA'- AAA'  •  />A'    because  p,p,p/  and 

AA//  ^^^  scalar, 

=  ^.  (/,A  •  /,//  .  AAO  •  4>  by  (38), 

=  L^L^L^ .  Zo ,  which  was  to  be  proved. 

Product  of  Three  Sects. — The  method  has  just  been  indi- 
cated, but  we  may  also  proceed  thus :  Let  the  lines  be 
Z„,  Z, ,  Z,,  and  let  A'A*  A  be  their  common  points.  Take 
scalars  n^^  n^    71^  so  that  Z^  =  ^^PiP^,  etc.,  then 

AAA  =  «o«i«,  -AA  'PJ.'PJ^  =  —  ^onrn,.p,pj,p,.p,p, 

=  -  «o«,«,  •  A/.A  •  A/0/1  =  n,n,nlp,p,p^\  (40) 

*  Grassmann  applies  the  terms  "eingewandt "  and  "  regressiv  "  to  a  prod- 
uct of  this  kind,  the  first  term  being  used  in  the  Ausdehnungslehre  of  1844, 
and  the  second  in  that  of  1862.  See  Chapter  3  of  the  first,  and  Chapter  3, 
Art.  94,  of  the  second. 


396  grassmann's  space  analysis.  [Chap.  VIII. 

Product  of  a  Point  and  Two  Sects. — Let/  be  any  point  and 
let  Zj  and  Z,  be  as  in  (38) ;  then 

/AA  =/.AA-AA  =/.AAA-A  =AAA-/A-     (40 

It  has  been  here  assumed  that/ZjZj  =/  .  L^L^.  The  prod- 
uct is  not  associative,  for  pL^  .  Z,  is  the  line  L^  times  the 
scalar /Zj,  a  different  meaning  from  that  assigned  in  (41).  As 
a  rule,  to  avoid  ambiguity,  the  grouping  of  such  products  will 
be  indicated  by  dots. 

Product  of  Two  Parallel  Sects. — Let  them  be/^e  and  np^e\ 
then,  as  in  (38), 

p^e .  np^e  =  n  .p.e  .p^6  =  n.  ep^  .ep^  =  n.  ep,p^ .  e,       (42) 
that  is,  a  scalar  times  the  common  point  at  00 . 

Addition  and  Subtraction  of  Sects. — Let  Z,  and  Z,  be  two 
sects,  /„  their  common  point,  and  /,  and  p^  so  taken  that 
A=AA»  A=AA;  then 

L,  +  A  =p,p,  -\-P.P.  =PSP.  +A)  =  2AA  (43) 

/  being  the  mean  of/,  and/,;  hence  the  sum  is  that  diagonal 
of  the  parallelogram  which  passes  through  /„.     Also 

^,  -  A  =  A(A  -A),  (44) 

so  that  the  difference  of  the  two  passes  also  through /„  and  is 
parallel  to  the  other  diagonal  of  the  parallelogram  determined 
by  Z,  and  Z,. 

If  the  two  sects  are  parallel  let  them  be  «,/je  and  n^p^e: 
then 

^./,e  +  nj^e  =  {nj,  +  n^p^e  =  {n,  +  «,)/e,,         (45) 

so  that  the  sum  is  a  sect  parallel  to  each  of  them,  having  a 
length  equal  to  the  sum  of  their  lengths,  and  at  distances  from 
them  inversely  proportional  to  their  lengths. 

If  «3  =  —  n^  the  two  sects  are  oppositely  directed  and  of 
equal  length,  and  the  sum  is 

«.(A€  -  Ae)  =  «,(A  -  A)e,  (46) 

which,  being  the  product  of  two  vectors,  is  a  scalar  area. 


Art.  7.]  PLANIMETRIC    PRODUCTS.  397 

Consider  next  ;^  sects /,e,  ,/,e,,  .  .  ./„e„,  and  let  e^  be  some 
arbitrarily  chosen  point;  then 

i/e  E  ^  Je  -  ^oie  +  |/e  =  ^„ie  +  i(/  -  ^Je.  (47) 

The  second  term  of  the  third  member  of  this  equation,  being  a 
sum  of  double  vector  products,  that  is,  a  sum  of  areas,  is  itself 
an  area,  and  is  equal  to  the  product  of  any  two  non-parallel  vec- 
tors of  suitable  lengths.  Therefore,  a  and  /?  being  such  vec- 
tors, write  ^e  =  a  and  2{p  —  e^)  =  a/3.  Hence  (47)  become 
^/e  =  e^a  +  a^  =  (e,  -  /3)a.  (48) 

Let  g  be  some  point  on  the  line  ^/e ;  then 
g2p€  =  0  =  qe^a  +  qap  =  qe^a  +  a/?, 
t>y  (37)»  hence     qe^a  =z  —  aft  =  fta. 

The  figure  presents  the  geometrical  mean- 
ing of  the  equation,  and  hence  it  appears  that 
qa{=2pe)  is  at  a  perpendicular  distance  from  e^ 

<?o  of 

aft       ^{p  -  ^„)6 

7^  =  — 7^— •  (49) 

It  is  easily  seen  that  a  sect  possesses  the  exact  geometrical 
properties  of  a  force,  namely,  magnitude,  direction,  and  position, 
and  the  discussion  of  the  summation  of  sects  which  has  just 
been  given  corresponds  completely  to  the  discussion  of  the  re- 
sultant of  a  system  of  forces  in  a  plane.  In  this  algebra,  then, 
the  resultant  of  any  system  of  forces  is  simply  their  sum,  and 
this  will  be  found  hereafter  to  be  equally  true  in  three-dimen- 
sional space.  The  expression  in  (46)  corresponds  to  a  couple, 
as  does  also  the  y,{p  —  e^e  of  (47);  and  this  equation  proves 
the  proposition  that  any  system  of  forces  in  a  plane  is  equiva- 
lent to  a  single  force  acting  at  an  arbitrary  point,  e^y  and  a 
couple.  Equation  (49)  gives  the  distance  of  the  resultant  from 
this  arbitrary  point. 

Exercise  7. — To  find  x.y^  z  from  the  scalar  equations 
^.^+^1  y-\r  c^^  =  < »     ^Q^  +  biy^c^^  =  ^> »     a^x-^b,y-{-  c,z  —  ^,. 


A      / 

/    \          / 

/ 

/ 

/ 

\/ 

/ 

— ^ 

398  grassmann's  space  analysis.  [Chap.  VIII, 

Multiply  the  equations  by/i,/^,  and /j  respectively,  and 
add ;  hence 

3  3  3  3 

x^ap  +  y'^bp  4-  z^cp  =  ^dp. 
1  111 

Now   '2ap,   '^bpj  etc.,  are  points  :   multiply  the  equation  just 
written  by  ^ap.  '2bp  ;  thus 

z:2ap  .  :2bp2cp  =  '^ap  .  :2bp .  :2dp, 
because  'Eap  .  '2ap  =  o,  etc.;  therefore 

2=:Eap.:^bp.2dp-^^ap.:^bp^cp  =  [a,,  b,,  d,-\^[a,,  b,,  c,-\, 
a  very  simple  proof  of  the  determinant  solution.     Of  course 
X  and  y  will  be  found  by  multiplying  by  the  other  pairs  of 
points. 

Exercise  8. — Forces  are  represented 
by  given  multiples  of  the  sides  of  a  par- 
allelogram ;  determine  their  resultant. 

Let  the  parallelogram  be  double  the 
triangle  e/^e^,  and  the  forces 

^o^o^i  +  K^r{e.  —  O  +  K^l^.  -  ^i)  +  k,e,e,  =  :Spe 

=  {k,  +  k,)e,e,  +  {k,  +  k:)e,e,  +  {k,  +  k^e^e^. 

Multiply  by  e^e^  to  find  where  the   resultant   cuts  this  line  ; 
then 

or  ^/i  cuts  the  resultant  at  the  point 

\(k,  +  k:^e,  -  {K  +  z^,)'.]  -  {K  -  k^). 
Similarly  the  resultant  cuts  the  other  sides  of  the  reference 
triangle  at  \iK  +  ^s)^,  -  i.K  +  ^.)^x]  -  {K  ^K-K-  k^  and 
at  [(/&„  +  k:)e,  -  {k,  +  K)e;\  -  {k^  -  K\ 

Suppose  k^=^  k^=  k^  =  k^',  then  each  of  the  three  points 
just  found  recedes  to  infinity ;  but  in  this  case  ^pe  reduces  to 
^kle.e,  +  e,e^  +  e^e,)  =  2kXe,  —  ^„)(^,  —  ^),  and  the  system  is 
equivalent  to  a  couple. 

Prob.  II.  Construct  the  resultant  of  Exercise  8  when  k^=  i, 
k,=  2,  k^—  3,  k^=  4;  when  k^=  i,  k^=  —  2,  k^=  3,  k^=  —  4;  when 
-^0  =  3>  ^1  =  ^3  —  2,  ^a  —  ^5  ^"^  when  ^^  =  ^,  =  i,  ^^  z=  /^^  =  —  2. 


■Art.  8.]  THE    COMPLEMENT.  399 

Prob.  12.  There  are  given  n  points  p^  .  .  ./„;  to  find  a  point  e 
such  that  forces  represented  by  the  sects  ep^ ,  ep^ ,  etc.,  shall  be  in 

equilibrium.     (The  equation  of  equilibrium  is  ^ep  =  e^p  =  -ep  =  o. 

Hence  e  coincides  with  the  mean   point  of  the/'s.) 

Prob.  13.  If  a  harmonic  range  ^i,/,  e^,  p'  be  given,  together  with 
■some  point  e^  not  coUinear  with  these  points,  show  that 

e,e,p  .  e.e^p'  =  -  e,pe, .  e,/e,. 
(Let  p  =  w/j  +  m^e^  and  p'  =  w/,  —  m^e^ ,    as    in    Exercise   2  of 
Art.  3.) 

Prob.  14.  Show  that  the  relation  of  Prob.  13  holds  for  any  four 
points  whatever  taken  respectively  on  the  four  lines  ^/, ,  e^p,  e^e^ , 
<?„/'.  If  the  four  points  are  all  at  the  same  distance  from  ^^ ,  show 
that  the  areas  e^e^p^  etc.,  become  proportional  to  the  sines  of  the 
angles  between  e^e^  and  e^p^  etc. 


Art.  8.    The  Complement.* 

Taking  point  reference  systems,  or  unit  normal  vector  ref- 
erence systems,  as  in  Art.  5,  the  product  of  the  reference  units 
taken  in  order  being  in  any  case  unity,  the  complement  of  any 
reference  unit  is  the  product  of  all  the  others  so  taken  that 
the  unit  times  its  complement  is  unity. 

To  find  the  complements  of  quantities  other  than  reference 
units  the  following  properties  are  assumed : 

{a)  The  complement  of  a  product  is  equal  to  the  product 
of  the  complements  of  its  factors. 

{b)  The  complement  of  a  sum  is  equal  to  the  sum  of  the 
complements  of  the  terms  added  together. 

(c)  The  complement  of  a  scalar  quantity  is  the  scalar  itself. 

Considering  now  the  point  system  in  plane  space  e^,  ^,,  e^ 
with  the  constant  condition  ^0^/,=  i,  the  sides  of  the  refer- 
ence triangle  taken  in  order  are  the  complements  of  the  oppo- 
site vertices,  and  vice  versa. 

The  complement  of  a  quantity  is  indicated  by  a  vertical 
line,  as  \p,  read,  complement  of/. 

*  See  Ausdehnungslehre  of  1862,  Art.  89. 


400  grassmann's  space  analysis.  [Chap.  VIII. 

Thus  ko  =  ^A'  ki^,  =l(ko)  =  ^o» 

ki  =  ^A>       k,^o  =  Kki)  =  ^i. 

k»=^o^x>        k.^x=l(ka)  =  ^,- 
For  ^J^„  =  ^o^/a  =  I,  which  agrees  with  the  definition  ; 

I  ^/,  =  kx  •  k,  =  ^»^.  •  ^0^1  =  -  ^0^,  •  ^0^,  =  -  ^o^a^x  •  ^0  =  ^0,  by  (^> 
and  (38) ; 

I  ^o^/«=  ko  •  k,  •  k,  =  ^i^«  •  ^.^0 .  ^0^1  =  (-^o^/.)'  =  I  =e,e,e^,  which 
agrees  with  (^) ;  ^^  ki  =  ^o^A  =  o  =  ^0  k»  =  ^x  ka- 

Next  take  any  point  p^  =  ^7^,  and  we  have,  by  (^), 

0 

l/,=iY|^=/../,+/.^,..+/,^.^.=/y.4g  -  gg  -  g  =  z,  (50) 

Thus  the  complement  of  a  point  is  a  Hne,*  which  may  be 
easily  constructed  by  the  fourth  member  of  (50),  which  ex- 
presses this  line  as  the  product  of  the  points  in  which  it  cuts 
the  sides  ^/,  and  e/^  of  the  reference  triangle.  Comparing 
this  equation  with  Ex.  3  in  Art.  4,  it  appears  that  \p^  above  is 

a  e 
related  to  the  point  ^  -v  as  the  line/^/,  of  Ex.  3  is  to  the  point 
0  / 

'2ne.  Hence  \p^  may  be  found  by  constructing  this  line  cor- 
responding  to  ^  -,  as  shown  in  the  figure  of  Ex.  3,  Art.  4. 

Again,  the  line  \p^  may  be  shown  to  be  the  anti-polar  of  p 
with  respect  to  an  ellipse  of  such  dimensions,  and  so  placed 
upon  ^//2  that,  with  reference  to  it,  each  side  of  the  reference 
triangle  is  the  anti-polar  of  the  opposite  vertex.*  From  this 
it  appears  that  complementary  relations  are  polar  reciprocal 

relations.     Take  any  point  p^  =  'Sine,  and  we  have 

0 

z=Slm  —  Sme  .Sl\e  =  p^\p^,  (51) 


*See  Hyde's  Directional  Calculus,  Arts.  41-43  and  121-123. 


Art.  8.J 


THE    COMPLEMENT, 


401 


so  that  this  product  is  commutative  about  the  complement 
sign,  and  scalar.  This  is  true  of  all  such  products  when  the 
quantities  on  each  side  of  the  complement  sign  are  of  the  same 
order  in  the  reference  units.  Take  for  instance  the  product 
AA 1/3/4-  This  is  scalar  because  \p^p^  is  a  point,  so  that  the 
whole  quantity  is  equivalent  to  a  triple-point  product ;  and  we 

havcA;^,  \p^,  =  1/ J>, .  AA  =  I  {P.P.  l/xA)  =  AA  i/iA»  by  (a)  and 
(c).  If,  however,  such  a  quantity  be  taken  as/j/^ .  |/,  it  is  neither 
scalar  nor  commutative  about  the  sign  |  ;  for,  \p^  being  a  Hne, 
the  product  is  that  of  two  lines,  that  is,  a  point,  and 

AA .  I A  =  - 1 A  'AA  =  -  I  (A  •  lAP.)-  (52) 

Such  products  as  we  have  just  been  considering  are  called 
by  Grassmann  "  inner  products,"  *  and  he  regards  the  sign  | 
as  a  multiplication  sign  for  this  sort  of  product.  Inasmuch, 
however,  as  these  products  do  not  differ  in  nature  from  those 
heretofore  considered,  it  appears  to  the  author  to  conduce  to 
simplicity  not  to  introduce  a  nomenclature  which  implies  a  new 
species  of  multiplication.  For  instance,/!^  will  be  treated  as 
the  combinatory  product  of  /  into  the  complement  of  ^,  and 
not  as  a  different  kind  of  product  oi  p  into  g. 

The  term  co-product  may  be  applied  to  such  expressions, 
regarded  as  an  abbreviation  merely,  after  the  analogy  of  cosine 
for  complement  of  the  sine. 

Consider  next  a  unit  normal  vector  system.  By  the  defini- 
tion we  have 

\^.=  u>  \h=  1(10=  -^i» 

because      zJZj  =  2jZ,  =  1, 

2,h.=  2,(- 0=-Vl=^^,=  I• 
Also,      ijz,=  — ZjZ,=0=zJZi. 


Next  let 


6,  =  m,i,  +  m^i,     and     €^  =  n,i^-{-  n^i^ ; 

*  Grassmann  (1S62),  Chapter  4. 


402  grassmann's  space  analysis.  [Chap.  VIII. 

then,  by  {b)  and  (<;), 

\e^=m,\i,+m^\i^  =  m,i^  -  m,i^.  (53) 

By  the  figure  it  is  evident  that  |  e,  is  a  vector  of  the  same 
length  as  e^  and  perpendicular  to  it,  or,  in  other  words,  taking 
the  complement  of  a  vector  in  plane  space  rotates  it  positively 
through  90°. 

The  co-product  e,  |  e^  is  the  area  of  the  parallelogram,  two 
of  whose  sides  are  e,  and  |  e,  drawn  outwards  from  a  point ;  if 
Cj  is  parallel  to  |  e, ,  this  area  vanishes,  or  e,  |  e,  =  o ;  but,  since 
1 62  is  perpendicular  to  e, ,  e^  must  in  this  case  be  perpendicular 
to  e^ ;  hence  the  equation 

eje,  =  0  (54) 

is  the  condition  that  two  vectors  e^  and  e^  shall  be  perpendicu- 
lar  to  each  other. 

The  co-product  e,  |  e^ ,  which  will  usually  be  written  e^^,  and 
called  the  co-square  of  e, ,  is  the  area  of  a  square  each  of 
whose  sides  has  the  length  Te^ ;  hence 

T6,=V'^,=V^-  (55) 

Let  a^  and  a^  be  the  angles  between  i^  and  e^  and  between 
z,  and  e,  respectively,  as  in  the  figure.     Then 

e^e,  =  m^n^  -  m^n^  =  Te.Te,  sin  (a,  —  a  J,  (56) 

the  third  member  being  the  ordinary  expression  for  the  area  of 
the  parallelogram  e^e^.     Also 

=  m^n,  -\-  m^n^  =  Te,  Te^  cos  {a,  —  a^),         (57) 

the  last  member  being  found  as    before,   remembering  that 
sin  (90°  -{-  a^—  a^  =  cos  {a^  —  a^). 

If  in  (57)  we  let  e^  =  e^,  whence  n^  =  m^  and  n^  =  m^,  we 
have 


Te,  =  ef.  =  Vm:  -+  <.  (58) 

li  Te,  =  T€^=  I,  then  m^  —  cos  a^,  m^  =  sin  a^,n,  =  cos  «'„ 
n^  =  sin  a^ ,  and  equations  (56)  and  (57)  give  the  ordinary  trigo- 
nometrical formulas  s{n{a^  —  a,)  =  sin  a^  cos  a,  —  cos  a^  sin  a^, 


Art.  8.]  THE  complement.  403 

and  cos  {a^  —  a^)  =  cos  a^  cos  a^  +  sin  a^  sin  or, .     Squaring  and 
adding  (56)  and  (57),  there  results 

re, .  r 'e,  =  e,i6,!  =  (e.e,)'  +  (e,  1 6,)^  (59) 

Attention  is  called  to  the  fact,  which  the  student  may  have 
already  noticed,  that  such  an  equation  as  AB  =  AC,  in  which 
AB  and  ^(7  are  combinatory  products,  does  not,  in  general, 
imply  that  B=Cf  for  the  reason  that  the  equation  A{B—C)=o 
can  usually  be  satisfied  without  either  factor  being  itself  zero. 
Thus  pL,  =P^i  rneans  simply  that  the  two  quantities  which 
are  equated  have  the  same  magnitude  and  sign,  which  permits 
Z,  to  have  an  infinity  of  lengths  and  positions,  when  p  and  L^ 
are  given.  The  equation p^p^  —  PxP%i  ^^PiiPi  —  A)  =  O,/^  and 
/g  being  unit  points,  impHes,  however,  that/,  =A>  unless  p,  is 
at  00 ,  that  is,  a  vector. 

Exercise  9. — A  triangle  whose  sides  are  of  constant  length 
moves  so  that  two  of  its  vertices  remain  on  two  fixed  lines : 
find  the  locus  of  the  other  vertex. 

Let  e^e,  and  e^e^  be  the  two  fixed  lines, 
and  pp'p"  the  triangle.  Let  pe  be  per- 
pendicular to  p'p"y  /'  —  ^0  =  ^^i  and 
P"  —  ^0  =  y^', ;  then  p"  —p'=  ye^  —  xe, , 
T{ye^  —  xe, )  z=  c  =  constant,  by  the  con- 
ditions. Also,  Tp^e  =  constant  =  mCy 
say,  and  Tep  =  constant  =  nCj  say.     Hence 

y€    —  X€ 

e-p'  =  Tp'e .  U(e  -/)  =  mc  .  ^^^^^  _  ^^^  =  m(ye,  -  xe,), 

and  similarly p  —  e  =  n\ {ys,  —  xe,).     Therefore 

p  —  e,  =  p  =  x€,  +  m{ye,  —  xe,)  +  n  \  {ye,  —  xe,\ 
an  equation  which,  with  the  condition  T{ye,  —  xe^  =  ^,  or 

jj/V,^  -  2xye,  I  e,  +  x'e,^  =  c\ 
determines  the  locus  to  be  a  second-degree  curve,  which  must 
in  fact  be  an  ellipse,  since  it  can  have  no  points  at  infinity. 
Let  us  rearrange  the  equation  in  p  thus  : 

p  =  ;ir[( I  -  m)e,  —  ^ |  e J  +  y\me^  +  n\e,]  =  xe  +  ye\  say. 


404 


GRASSMANN  S    SPACE    ANALYSIS. 


[Chap.  VIIU 


SO  that  e  =  (i  —  m)e^  —  n\e^  and  e'  =  me^  +  ^1^2;  then  multi- 
ply successively  into  e  and  e'\  therefore  pe  =  ye' e  and 
pe'  =  xee' .  Substituting  these  values  of  x  and  j^  in  the  equa- 
tion of  condition,  we  have 

el .  (pe)"  +  2e,  |  e, .  pe  .  pe'  -f  e^P^)^  =  c^ee'Y, 

a  scalar  equation  of  the  second  degree  in  p. 

Exercise  lo. — There  is  given  an  irregular  polygon  of  n 
sides :  show  that  if  forces  act  at  the  middle  points  of  these 
sides,  proportional  to  them  in  magnitude,  and  directed  all  out- 
ward or  else  all  inward,  these  forces  will  be  in  equilibrium. 

Let  e^  be  a  vertex  of  the  polygon,  and  let  2e^,  2e^,.  . .  2e„ 
represent  its  sides  in  magnitude  and  direction.  Then  the  mid- 
dle points  will  be  ^o+^i>  ^o+2e, -f-e^,  etc.,  and,  using  the 
complement  in  a  vector  system,  we  have 

^/e  =  {e,+e,)  \  e,+(^„-f  2e,+e,)  1  e,+(^„+2e,+2e,+e3)  |  e,+  . . . . 

+  (^0  +  2e,  +  .  . .  +  2e«_i  +  e„)| e„. 


=  ^„ 


:2'e  +  ^V-  +  2e, 


2e  +  2e, 


:S'e+...  +  2e„_i|e, 


n  /  M       \  2 

2'e+  f-2e]~=  o,  which  was  to  be  proved. 

Exercise  1 1. — A  line  passes  through  a  fixed  point  and  cuts 
two  fixed  lines ;  at  the  points  of  inter- 
section perpendiculars  to  the  fixed  lines 
are  erected  ;  find  the  locus  of  the  inter- 
section of  these  perpendiculars. 

Let  the  fixed  lines  be  ^^ej  and  e^e^ , 
and  the  fixed  point  e^  -{-  e,;  the  moving 
line  cuts  the  fixed  lines  in  />'  and  p", 
at  which  points  perpendiculars  are 
erected  meeting  in  p. 

Let/  —  e,  =  p,/  —  e,  =  xe,,p"-  e,  —  ye^ ,  Te,  =Te^  =i; 

then  p  =  xe^-{-x' \e,  —  ye^-\-y' \e^  ,  whence  p|e,=  ;ir  and  p\e^=  y. 


Art.  9.]  EQUATIONS    OF    CONDITION,    AND    FORMULAS.  405 

Also,  since  e^  +  e^,  p' ,  p"  are  collinear  points, 

{xe^  -  e^){ye^  —  e,)  =  o  =  xye,€^-\- ye^e^  +  X6^e^\ 
or,  substituting  values  of  x  and  y, 

p|  e, .  p|  e, .  e,e,  +  p|  e, .  e^e,  +  /o|  c,  .  e^e^  =  O, 
an  equation  of  the  second  degree  in  /o,  and  hence  representing 
a  conic. 

Prob.  15.  If  ^,  <^,  ^  are  the  lengths  of  the  sides  of  a  triangle,  prove 
the  formula  a"  =  d"^  -\-  c'^  —  2bc  cos  A^  by  taking  vectors  e,,  e,,  and 
€a  —  ei  equal  to  the  respective  sides. 

Prob.  16.  If  ^„€j  and  e^ei  are  two  unit  lines,  show  that  the  vec- 
tor perpendicular  from  e^  on  the  line  (e^  +  ae^{e^  +  be^  is 

*       - .  I  {bet.  —  «ej),  of  which  the  length  is  — -r- — ^—^ — r^.    From 


{be^-aey-''^  '"  °  Tibe^^-ae^ 

this  derive  the  Cartesian  expression  for  the  perpendicular  from  the 
origin  upon  a  straight  line  in  oblique  coordinates, 
ub  sin  ct?  -7-  (^'  +  /^*  —  2ab  cos  cj?)^,  c»  being  angle  between  the  axes. 
Prob.  17.  If  three  points,  me^  -\-  ne^,  me^  +  ne^,  me^  -f-  ne^,  be 
taken  on  the  sides  of  the  reference  triangle,  then  the  sides  of  the 
complementary  triangle,  |  {me^  -\-  ne^),  etc.,  will  be  respectively  paral- 
lel to  the  corresponding  sides  of  the  triangle  formed  by  the  assumed 
points  {me J  +  ^^J,  (^<^2  +  ^i^o)y  etc. 

Art.  9.    Equations  of  Condition,  and  Formulas. 

Several  equations  of  condition  are  placed  here  together  for 
convenient  reference  :  some  have  been  already  given  ;  others 
follow  from  the  results  of  Arts.  7  and  8.     When  we  have 

L,L,  =  o, 


P.P.  =  o, ) 


or        n,L,  +  n,L,  =  o, 
the  two  lines  coincide; 

L.L^L,  =  o, 

3 
or  2nL  =  o, 


(60) 


(61) 


or 

the  two  points  coincide  ; 
AAA  =  0,1 
or  2np  =  o,  I 

the  three  points  are  collinear ; 

€,e,  =  o,     or     n,€,  +  ^q^j  =  o,  (62) 

the  two  vectors  are  parallel  (points  at  infinity  coincide); 

e.  I  e,  r=:  o,  (63) 


J 
the  three  lines  are  confluent. 


406  grassmann's  space  analysis.  [Chap.  VIII. 

the  two  vectors  are  perpendicular ; 


either  point  lies  on  the  com- 
plementary line  of  the  other. 


AIA  =  o,  (64) 

either  line  passes  through  the 
complementary  point  of  the 
other. 


If  we  write  the  equation 

P  =  ^1^1  +  ^36,, 
^jr^e^  is  the  projection  of  p  on  ei  parallel  to  e^,  and  ;irje,  is  the 
projection  of  p  on  e,  parallel  to  e^.     Multiply  both  sides  of  the 
equation   into  e^;    therefore  pe,  —  ^^e^e^,  or   ;r,  =  pe,  -r-  e^e^^ 
Similarly,  multiplying  into  e,,  we  have  pe,  =  -^^e^e^f  or 
x^  =  pej  -r-  €36,,  whence 

The  two  terms  of  the  second  member  of  (65)  are  therefore 
the  projections  of  p  on  e,  parallel  to  e,,  and  on  e^  parallel  to  e,, 
respectively.* 

Let  Gj  and  e,  be  unit  normal  vectors,  say,  i  and  \i;  then  (65) 

becomes 

p  =  z.p|z—  \i,  pt  =  t.p\i.+  ip.\i;  (66) 

or,  if  Zj  and  i,  be  used  instead  of  i  and  |  z, 

p=  i,.p\z,  +  i,.p\i^.  {67) 

Again,  in  (65)  let  p  =  63,  clear  of  fractions,  and  transpose ; 
therefore 

e.e^ .  €3  +  €263 .  e^  +  e^e, .  e,  =  o,  (68) 

a  symmetrical  relation  between  any  three  directions  in  plane 
space.     Let  fe^  =  Te^  =  Te^  =  i,  and  multiply  (68)  into  leg, 

thus  e^e^  +  €,€3  .e^\€,-{-  €,e,.€^\e^  =  O,  (69) 

which  is  equivalent  to 

sin  {a  ±  /3)  =  sin  a  cos  /3  ±  cos  a  sin  y^, 

the  upper  or  lower  sign  corresponding  to  the  case  when  e^  is 

*  Grassmann  (1844),  Chapter  5  (1862), /\rt.  129.  Hyde's  Directional  Calcu- 
lus, Arts.  46  and  47. 


EQUATIONS    OF    CONDITION,    AND    FORMULAS. 


407 


Art.  9.] 

between  e^  and  e^,  or  outside,  respectively.  Writing  in  (69)  ]  e^, 
instead  of  e^,  we  have 

6,\€,  —  €,\6,  .  e,\€,  +  636,  .  6,63  =0,  (70) 

which  gives  the  cos  {a  ±  /3).  These  formulas  being  for  any 
three  directions  in  plane  space,  are  independent  of  the  magni- 
tude of  the  angles  involved. 

There  is  given  below  a  set  of  formulas  for  points  and  lines, 
arranged  in  complementary  pairs,  and  all  placed  together  for 
convenient  reference,  the  derivation  of  them  following  after. 

/=(AAA)''[A  -A^iA  +  A  -/AA  +  A  -/AAl. 


/=(AAA)"TIAA-/IA  +i  AA./IA+  I  AA-Zl A], 

L 


,-/lA]J 


(71) 
(72) 


(73) 


AA-AA  =  -  A  •  AAA  +  A  -AAA 

=     A-AAA-A-AAA*      I 

A  Al^ 

_    L,\M,  L 
~    L,\M,  L 

h  PM'i 

The  complementary  formula  to  ijj^  is  not  given,  but  may 
be  obtained  by  putting  Z's  and  M's  for  /'s  and  qs. 

Derivation  of  Equations  (7i)-(77). — Equation  (71).  Write 
/  =  ^oA  +  -^lA  +  ^3A»  ^"d  multiply  this  equation  by  p^p^ ; 
then  p,p,p  =  -^oAAA.     or     -^0  =  MA  ^PoPJ.' 

Multiplying  similarly  by  p  /,  and  by /„/>,,  we  find 
^1  =  /AA  -^PJ^P.  and  .r,  =  pp,p,  ^p.p^P,-     The  substitu- 


AA.|^.= 

— 

A  AU 

A  AU 

^1 

,      ^.^ 

■,\M,^ 

— 

Z,   L,\M, 

> 

(74) 

Al^x^.= 

1^1  Aki 

1^.   Aka 

> 

(75) 

AA|^.^a  = 

Aki  Ak. 
Al^i  Al^. 

,    L,L,\M,M,= 

LAM,  LAM, 
L,\M,  L,\M, 

.(76) 

PM.PM.  Al?. 

/o/aA  •  ^0^:^,  = 

PM.PAq^PM^ 

(77) 

PM.P 

,k, 

pm. 

408 


grassmann's  space  analysis. 


[Chap.  VIII, 


tion  of  these  values  gives  the  first  of  (71),  and  the  second  is 
similarly  obtained  or  may  be  found  by  simply  putting  Z's  for 
/'s  in  the  first. 

Equation  (72).  Write/  =  x^  \p,  A  +  ^ J  AA  +  -^2 1  A /n  and 
multiply  into  |A  i  thus/|A  —  ^^P^PiPi-  Find  in  the  same  way 
values  of  x^  and  x^,  and  substitute. 

Equation  (73).  Write /j  A  -AA  =^  -^/i  +7A'  and  multiply 
by /A  ;  therefore /A  -AA  •  A  A  =  ^/AA»  or,  by  Art.  23, 
A//i  •  AAA  =  -^/A  A  =  -  •^'AA^ii  or,  ;«;  =  -  AAA-  Multiply- 
ing by/A  ^^'^  ^"<^  J^==AAA»  and  on  substituting  obtain  the 
first  of  (73).  For  the  second  put  /,A  -AA  —  -^A  +  JA»  and 
proceed  in  a  similar  way. 

Equation  (74).  In  the  first  of  (73)  put  />gA  =  |^i. 

Equation  (75).   In  the  fourth  of  (73)  put 

AA  =A»  A=  ku  A  =ka. 

Equation  (76).  Multiply  (75)  by /j. 

Equation  {^'jj).  In  the  first  of  (72)  put  q.^  for  A  and  multiply 
byAA^. -^o^i;  then 
A/.A  •  ^0^,^.  =  ^o^>  l/iA  •  ^.  I/0+  ^0^1  lAA  •  ^.  l/,+  ^0^1  \P.P^  •  ^.  lA 


Ak^ 


A  1^0   A  1^1 
Al^o  Al^i 


+Ak. 


+  Ak. 


Al^o  Al^. 
Al^o  AiVi 


Al^o   Ai^, 

Ai^o  A'^i' 

by  i^'jG),  which  is  equivalent  to  the  third  order  determinant  of 

equation  {jf)-'^ 

Exercise  12. — To  show  the  product  of  two  determinants  as 

a  determinant  of  the  same  order. 
2 

0 
then  A/,  A  =  Uoy  ^«n  «J>  ^0^:^,  =  [K  f^v  "^J  ;  also 
/o  ko  =  ^0^0  +  A^x  +  hK  PA^o  =  ^0^0  +  ^.\  +  ^^X>  etc.    Sub- 
stituting these  values  in  {jy),  we  have  the  required  result.     A 

solution  may  also  be  obtained  directly  without  the  use  of  {77)» 

2 

Let  the  ^'s  be  as  above,  but  write  A  =  ^h^Px  ~  ^^^^»  A  =  ^M- 

0 

Then 

p^p^p^=:2lq ,:2mq.:2nq=\l^,  m^,n^q^q^q^^\l^,  m,,  «J[A„,  //,,  rj. 
*  Grassmann  (1862),  Art.  173, 


Art.  9.]  EQUATIONS    OF    CONDITION,    AND    FORMULAS.  409 

Also  p,  =  l,:2\e  + 1^2  ^e  +  i;2ve 

with  similar  values  for/^  and/^*  which  on  being  substituted  in 
pQp\Pi  §^^^  ^^^  result.  Equation  {yy),  however,  exhibits  the 
product  in  a  very  compact,  symmetrical,  and  easily  remembered 
form."^ 

Exercise  13. — Show  that  the  sides /,/,, /^A^  A/^i  of  the  tri- 
angle/,/j/3  cut  the  corresponding  sides  |/,,  |/j,  I/,  of  the  com- 
plementary triangle  in  three  coUinear  points. 

The  three  points  of  intersection  are,  using  (74), 
/.A  .  I A  =  -  P.  -A  \P^+P.'P.  lA,  AA .  I A  =  -A  -A  I A  +A  -A  I  A, 
P.P^'\P^  =  -pz'Pi\P'^+Px'Pz\P^^  of  which  the  sum  is  zero, 
showing  that  the  points  are  collinear.  It  may  be  shown  in 
the  same  way  that  the  lines  joining  corresponding  vertices  are 
confluent. 

Exercise  14. — If  the  sides  of  a  triangle  pass  through  three 
fixed  points,  and  two  of  the  vertices 
slide  on  fixed  hnes,  find  the  locus  of 
the  other  vertex. 

Let  the  fixed  points  and  lines  be    ""^i 
p,,  /„  /3,  Z„  Z„  and  /,  /,  p"  the  .- 

vertices  of   the  triangle,   as   in    the         * 
figure.     Then  p'p^p"  -=  o\  p'    coin- 
cides  with  //, .  Zj    and/''   with  pp^,  L^\    hence    substituting 
(/'/^ .  Zj)A(^2-A/)  —  o,  the  equation  of  the  locus,  which,  being 
of  the  second  degree  in/,  is  that  of  a  conic. 

Prob.  18.  Show  that  if  the  three  fixed  points  of  the  last  exercise 
are  cohinear,  then  the  locus  of  /  breaks  up  into  two  straight  lines. 
Use  equation  (73). 

Prob.  19.  If  the  vertices  of  a  triangle  slide  on  three  fixed  lines, 
and  two  of  the  sides  pass  through  fixed  points,  find  the  envelope  of 
the  other  side.  (This  statement  is  reciprocally  related  to  that  of 
Exercise   14,  that  is,  lines  and  points  are  replaced  by  points  and 

*  These  methods  may  be  applied  to  determinants  of  any  order  by  using  a 
space  of  corresponding  order. 


4X0  grassmann's  space  analysis.  [Chap.  VIII. 

lines  respectively,  and  the  resulting  equation  will  be  an  equation  of 
the  second  order  in  Z,  a  variable  line.) 

Prob.  20.  Show  that  if  the  three  fixed  lines  of  Exercise  5  are 
confluent,  then  the  envelope  of  L  reduces  to  two  points  and  the  line 
joining  them. 

Art.  10.    Stereometric  Products. 

The  product  of  two  points  in  solid  space  is  the  same  as  in 
plane  space.     See  Art.  7. 

Product  of  Three  Points. — Any  three  points  determine  a 
plane,  and  also,  as  in  Art.  7,  an  area ;  hence /j/^A  is  a  plane-sect 
or  a  portion  of  the  plane  fixed  by  the  three  points  whose 
area  is  double  that  of  the  triangle  p^p^p^  It  may  be  shown,  in 
the  manner  used  in  Art.  7  for  the  sect,  that  no  plane-sect,  not 
in  this  plane,  can  be  equal  to  p^p^p^,  and  that  any  plane-sect  in 
this  plane  having  the  same  area  and  sign  will  be  equal  to/^^/*,.* 
Of  course /j/j/j  is  not  now  scalar. 

Product  of    Four  Points. — Any  four   non-coplanar   points 

determine  a  tetrahedron,  say 

^^'^\^  /^i/aAA'  ^"d  six  times  the  vol- 

Xv  "\     ume   of    this    tetrahedron    is 

>C  y'     taken    for   the    value   of    the 

\  y'  product,  because  this   is  the 

volume  of  the  parallelepiped 

generated  by  the  product /,/,/„ — i.e.  the  parallelogram Z^,/,, — 

when  it  moves  parallel  to  its  initial  position  from/,  to /^.     Let 

A  -A  =  e, A  -A  =  e^ A  -A  =  e'^  then 

/AAA  =  AAA^^'  =  PrP.e'e"  =  Aee'e''.  (78) 

333  3 

If/,  =^ke,p^  =  2/e,p^  =  ^me,p^  =  ^ne,  then 
000  0 

PiP.p.P^  =  ^ke^le^me^ne  =  \k,,  /,,  m^,  n,]  .  e.e^e^e^ ;     (79) 

from  which   it  appears  that  any  two  quadruple  products  of 

points  differ  from  each  other  only  by  a  scalar  factor,  that  is,  they 

differ  only  in  magnitude,  or  sign,  or  both  ;  hence  such  products 

are   themselves  scalar.f     If  /lAAA  =  o>   the   volume   of  the 

tetrahedron  vanishes,  so  that  the  four  points  are  coplanar. 

*  Grassmann  (1862),  Art.  255.  f  Grassmann  (1862),  Art.  263. 


Art.  10.]  STEREOMETRIC  PRODUCTS.  411 

Product  of  Two  Vectors. — The  two  vectors  determine  an 
area  as  in  Art.  7,  but  they  also  determine  now  a  plane  direc- 
tion, so  that  the  product  e^e^  is  a  plane-vector,  and  is  not  scalar 
as  in  plane  space.  Also,  e^e,  differs  from  />^e^e^  now  just  as  e 
differs  from  /e;  namely,  e^e,  has  a  definite  area  and  plane 
direction,  that  is,  toward  a  certain  line  at  infinity,  while  p.e^e^  is 
fixed  in  position  by  passing  through /j.  Equation  (37)  there- 
fore does  not  hold  in  solid  space. 

Product    of   Three   Vectors. — Three  vectors   determine  a 

parallelepiped  as  in  the  figure  above,  and  ee^e''  is   therefore 

the  volume  of  this   parallelepiped.     Any  other  triple   vector 

product  can  differ  from  this  only  in  magnitude  and  sign.     For 

let  e^e^e,  be  such  a  product,  and  write 

33  3 

e  =  x^e^  +  ^a^a+  -^3^3  =  ^^€t  e'  =  ^J^e,  e"  =  ^ze ;  then 


'^xe'^ye^ze 


(80) 


y.  y,  J3 

^j    z^    z^ 

so  that  the  two  products  only  differ  by  the  scalar  determinant 
factor.  Hence  the  product  of  three  vectors  must  be  itself  a 
scalar,  by  Art.  i.  Since,  then,  the  product  of  four  points  has 
precisely  the  same  signification  as  that  of  three  vectors,  we  may 
write 

/'.AAA  =P.ee'e''  =  ee'e''  =  (A  -  AXA  -  A)(A  -  A) 

=  AP.P*  -  /s/./.  +  A/./,  -  AP.P.'  (81) 

Thus  the  sum  of  the  plane-sects  forming  the  doubles  of  the 
faces  of  a  tetrahedron,  all  taken  positively  in  the  same  sense 
as  looked  at  from  outside  the  tetrahedron,  is  equal  to  the 
volume  of  the  tetrahedron.     Compare  equation  (37). 

If   ee'e'^  =  o,   the  volume   of  the  parallelepiped  vanishes, 
and  the  three  vectors  must  be  parallel  to  one  plane. 

Product  of  Two  Sects. — In  solid  space  two  sects  determine 
a  tetrahedron  of  which  they  are  opposite  edges.     Thus 

APJJ*  =  P.P. ' P.P.  =  L,L,  =  P. P.  •  /A  =  L,L,,  (82) 
so  that  the  stereometric  product  of  two  sects  is  commutative, 
and  has  the  same  meaning  as  that  of  four  points. 


>.  }  (^3) 


412  grassmann's  space  analysis.  [Chap.  VIII. 

Product  of  a  Sect  and  a  Plane-Sect. — Let  them  be  L  and 
P,  and  let  /„  be  their  common  point;  take  /i,A»A  so  that 
Z=/»oA  ^i^d  P=pop2pi'  ^  ^rid  P  evidently  determine  the 
point  /o,  and  also  the  parallelepiped  of  which  one  edge  is  L 
and  one  face  is  P,  so  that  the  product  should  be  made  up  of 
these  two  factors.     Hence  we  write 

^-P  =  P.P.  •  AAA  =  P.P.P.P. .  A; 

PL  =^PJ,P,  -P.P,  =P.P.P^P.  -A  =  LP. 

If  L  is  parallel  to  P,  A  is  at  infinity,  and,  replacing  it  by  e, 
(83)  becomes 

PL^LP^  ep, .  eA/3  =  e/), AA  .  e.  (84) 

Product  of  Two  Plane-Sects. — Let  them  be  P^  and  P^,  and 
let  L  be  their  intersection,  while  A  and  A  ^-^e  such  points  that 
P,  =  ZA  and  P,  =  ZA;  then  P,  and  P,  determine  the  line  L 
and  also  a  parallelepiped  of  which  they  are  two  adjacent  faces, 
and 

P,P,  =  Z/,  .  Z/,  =  Z/,A  .  Z  :=   -  P,/',  (85) 

If  P,  and  Pj  are  parallel,  L  is  at  infinity,  and  is  equivalent 
to  a  plane-vector,  say  to  77 ;  hence,  substituting  in  (84), 

P^P,  ==  -np, .  r^p,  =  r^pj, .  /;  =  -  P,P,.  (86) 

Product  of  Three  Plane-Sects.— By  (85)  and  (83)  this  must 
ibe  the  square  of  a  volume  times  the  common  point  of  the 
three  planes;  or,  if  A' A»  A>A  ^^  taken  in  such  manner  that 
P.  =PJ.P..  P.  =PoP.P..  p.  =P.PJ..  then 

p,p,P3  =023 .  031 .  012  =.  023 .  0123 .  01  =  {p,p,p,p,Y  -A ;  (87) 

the  suffixes  being  used  instead  of  the  corresponding  points. 
If  A  be  at  infinity,  the  three  planes  are  parallel  to  a  single  line, 
and  may  be  written  P,  =  ?2,eAA»  ^^c,  and  then  treated  as 
above. 

Product  of  Four  Plane-Sects."^ — Let  the  planes  be  P^ .  .  .  P3, 
and  let  A  •  •  •  A  be  the  four  common  points  of  the  planes  taken 
three  by  three,  n^  .  .  .  n^  may  be  so  taken  that  P^  =  «o/iAA> 
^tc. ;  then 

P,P^P^P,  =  n,n,n^n, .  123  .  230 .  301  .  012 

=  nori,n,n^{p,p,pj>,)\  (88) 

*  Grassmann  (1862),  Art.  300. 


Art.  10.]  STEREOMETRIC    PRODUCTS.  415 

Product  of  Two  Plane- Vectors. — Let  r/^  and  tf^  be  two  plane- 
vectors  or  lines  at  infinity ;  let  e  be  parallel  to  each  of  them, 
and  ej  and  e^  so  taken  that  7;,  =  ee,,  ?;,  =  ee,,  then 

VxV^  =  ee, .  ee,  =  ee.e, .  e  =  —  rf^t},,  (89) 

because  tf^  and  rj^  determine  a  common  direction  e,  and  a  paral- 
lelepiped of  which  three  conterminous  edges  are  equal  to 
e,  e,,  e,,  respectively. 

Product  of  Three  Plane-Vectors. — Take  e^,  e„  Cg  so  that 

VxV^V,  =  «  .  e,e,  .  e,e, .  e,e,  =  n^e.e^e^f,  (90) 

The  directions  e,  ...  eg  are  common  to  the  plane-vectors 
T^j  .  .  .  7;,  taken  two  by  two. 

Several  conditions  are  given  here  together  which  follow 
from  the  results  of  this  article. 

AA  =  o,  P.P.  =  o,  (91) 

Two  points  coincide.  Two  planes  coincide. 

/,/,/,  =  o,  P^P^P,  =  o,  (92) 

Three  points  collinear.  Three  planes  collinear. 

/>/,AA  =  /xA  .  AA  P.P^P^P.  =  P.P. .  P.P. 

=  Z,Z,  =  o,  .                        =  Z,Z,  =  o,        (93) 

Four  points  coplanar;  two  Four    planes    confluent;    two 

lines  intersect.  lines  intersect. 

€,e,  =  0,  ^x7,  =  o,                               (94> 

Vectors  parallel.  Plane-vectors  parallel. 

e^e^Cg  =  o,  VxV.Vi  =  O,                             (95) 

Three  vectors   parallel  to           Three  plane-vectors  parallel  to 

one  plane.  one  line. 

Sum  of  Two  Planes. — Let  them  be  P^  and  /!,,  let  Z  be  a 
sect  in  their  common  line,  and  take  p^  and/,  so  that  P,  =  Z/,, 
P^  =  Lp^ ;  then 

P,  +  P,  =  Z(A+A)=2ZA  .    (96) 

p  being  the  mean  of /^  and  p^.     Also 

P,-P,^L{p,-py,  (97) 

whence  the  sum  and  difference  are  the  diagonal  plane  through 
Z,  and  a  plane  through  L  parallel  to  the  diagonal  plane  which 
is  itself  parallel  to  Z,  of  the  parallelepiped  determined  by  /', 


414  grassmann's  space  analysis.  [Chap.  VIII. 

and  P,.  If  TP,  =  TP^,  P,  ±  P^  will  evidently  be  the  two 
bisecting  planes  of  the  angle  between  them.  The  bisecting 
planes  may  also  be  written 

Yp^±Yp^     or     PJP,±P,TP,.  (98) 

If  the  two  planes  are  parallel,  let  7;  be  a  plane-vector 
parallel  to  each  of  them,  that  is,  their  common  line  at  infinity, 
and  let/i  and  p^  be  points  in  the  respective  planes;  then  we 
may  write  P,  —  n^pj],  P^  —  n^p^rf,  whence 

P.+P.  =  in,p,-]rn^,)V  =  {n,-\-nS~pV'  (99) 

If  n^-\-n^=^  o,  this  becomes 

P.  +  P.=-n^{P.-pH  (100) 

the  product  of  a  vector  into  a  plane-vector  and  therefore  a 
scalar,  by  (80). 

Two  plane-vectors  may  be  added  similarly,  since  they  will 
have  a  common  direction,  namely,  that  of  the  vector  parallel 
to  both  of  them. 

Exercise  15. — If  two  tetrahedra  e^e^e^e^  and  e^e^e^e^  are  so 
situated  that  the  right  lines  through  the  pairs  of  corresponding 
vertices  all  meet  in  one  point,  then  will  the  corresponding  faces 
cut  each  other  in  four  coplanar  lines. 

The   given    conditions    are    equivalent    to    ^//  .  e^e^  =  o 

=    ^0^0'  .  ^.^/  =   ^0^0'  •  ^3^3'  =    ^i^i'  •  ^//  =   ^.//  -^3^3'    =   ^3^3'  .  ^^^/' 

Two  of  the  intersecting  lines  of  faces  are  e/^e^ .  e^e^e^  and 
^//s  •  ^x^^^-l->  ^"d,  if  these  intersect,  we  must  accordingly  have, 
by  (92),  012  .  o'i'2'  .  123  .  \'2'i'  =  o  =  012  .  123  .  o'\'2'  .  l'2'3 
=  0123.0' i'2'3'.  121^2',  the  last  factor  of  which  is  equivalent 
to  the  fourth  condition  above,  since  quadruple-point  products 
in  solid  space  are  associative.  Similarly  all  the  other  pairs 
of  intersections  may  be  treated. 

Exercise  16. — The  twelve  bisecting  planes  of  the  diedral 
angles  of  a  tetrahedron  fix  eight  points,  the  centers  of  the 
inscribed  and  escribed  spheres,  through  which  they  pass  six 
by  six. 

The  sum  and  difference  of  two  unit  planes  are  their  two 


Art.  10.]  STEREOMETRIC    PRODUCTS.  415 

bisecting  planes,  by  (97).  Let  the  tetrahedron  be  e^e^e^e^^  and 
let  the  double  areas  of  its  faces  be  A^  =  Te^e^e^,  etc.;  then  a 

6  €  6  €  €  6 

pair  of  bisecting  planes  will  be  -~-^  ±  -j^  or  e^e^{A^e^  ±  A^e^). 

The  pair  through  the  opposite  edge  will  be  e^e^A^e^  ±  A^e^). 
If  there  be  a  point  through  which  the  six  internal  bisecting 
planes  pass,  it  must  be  on  the  intersection  of  these  two  planes 

taken  with  the  upper  signs,  and  we  infer  by  symmetry  that  it 

3 
must  be  the  point  ^Ae.     Another  internal   bisecting  plane  is 

0 

e^eJ^A^e^  -\-  A^e^,  which  gives  zero  when  multiplied  into  '^Ae^ 
as  do  also  the  other  three. 

To  obtain  all  the  points  we  have  only  to  use  the  double 
signs,  so  that  they  are  ±  A^e^  ±  A^e^  ±  A^e^  ±  A^e^  This 
gives  eight  cases,  narnely, 

++++  -+++ 
+  +  +  -  +  +  — 
+  +  -+         +  —  + 

The  eight  apparent  cases  that  would  arise  by  changing  all  the 
signs  are  included  in  these  because  the  points  must  be  essen- 
tially positive.  Moreover,  no  positive  point  could  have  three 
negative  signs,  because  the  sum  of  any  three  faces  of  the  tetra- 
hedron must  be  greater  than  the  fourth  face.  It  will  be  found 
on  trial  that  six  of  the  bisecting  planes  will  pass  through 
2(  ±  Ae)  with  any  one  of  the  above  arrangements  of  sign. 

Prob.  21.  The  twelve  points  in  which  the  edges  of  a  tetrahedron 
are  cut  by  the  bisecting  planes  of  the  opposite  diedral  angles  fix 
eight  planes,  each  of  which  passes  through  six  of  them. 

Prob.  22.  The  centroid  of  the  faces  of  a  tetrahedron  coincides 
with  the  center  of  the  sphere  inscribed  within  the  tetrahedron 
whose  vertices  are  the  centroids  of  the  respective  faces  of  the  first 
tetrahedron. 

Prob.  23.  If  any  plane  be  passed  through  the  middle  points  of 
two  opposite  edges  of  a  tetrahedron,  it  will  divide  the  volume  of  the 
tetrahedron  into  two  equal  parts. 


416  grassmann's  space  analysis.  [Chap.  VIII. 

Art.  11.    The  Complement  in  Solid  Space. 

According  to  the  definitions  of  Art.  8  the  complementary 
relations  in  a  unit  normal  vector  system  are  as  follows : 

■  hi  =  hh^  \hh  =  1(10  =  ^ 

1(10  =  ^ 

1(10  =  ^ 


1,1. 


(lOl) 


Let  e  =  2h ;  then 


\e 


h\h  +  hh\  +  hh\  =  -j^h^  -  K^^{K\ 


/,0»  (102) 


-^.''%«3 


so  that  |e  is  a  plane-vector.     The  figure,  which  is  drawn    in 

isometric  projection,  shows 
that  the  two  vectors  l^i^  —  l^i^ 
and  /jZg  —  l^t^,  whose  prod- 
uct is  /, .  I  e,  are  both  perpen- 
dicular to  e ;  for  the  first  is 
perpendicular  to  l^i^  -|-  l^i^, 
which  is  the  orthogonal  pro- 
jection of  eupon  z,z3,and  to 
Zg,  and  therefore  is  also  per- 
pendicular to  e,  while  the 
second  is  perpendicular  to  /,t,  +  l^i^  and  to  z.^,  and  therefore 
to  e.  Hence  |  e  is  a  plane-vector  perpendicular  to  e  ;  and,  since 
1(1  e)=  e,  the  converse  is  also  true,  i.e.  the  complement  of  a 
plane-vector  is  a  line-vector  normal  to  it. 

The  figure  shows  that  e  is  equal  to  the  vector  diagonal  of 
the  rectangular  parallelepiped  whose  edges  have  the  lengths 
/„  /„  /g,  hence  

7^6=1//,'  +  //  +  /;.  (103) 

Multiply  equation  (102)  by  e;  therefore 

e\e  =  {1,1,  +  1,1^  +  l,i,){l,i,i,  +  l^z^i,  +  /,^0 

=  i:+i:  +  i:=T^e  =  d,  (I04> 

so  that  the  co-square  of  a  vector  is  equal  to  the  square  of  its 
tensor.  The  product  e|e  is  that  of  a  vector  e  into  a  plane- 
vector  perpendicular  to  it,  as  has  just  been  shown  ;  it  is  there- 


Art.  11.] 


THE   COMPLEMENT    IN    SOLID    SPACE. 


417 


i-A 


fore  a  volume  which  is  equivalent  to  Te .  T\e\  hence,  by  (104), 
el  e  =  Te  ,  T\e—  T'^e,  or  Te  =■  T\  e.  Hence,  the  complement 
of  a  vector  in  solid  space  is  a  plane-vector  perpendicular  to  it 
and  having  the  same  tensor,  or  numerical  measure  of  magni- 
tude.* 

3 
Let  a  second  vector  be  e'  =2^mi ;  then 

1 

e  I  e'  =  l,m,  +  l^rn^  +  l^m,  =  e'\e,  (105) 

Now  e|e',  being  the  product  of  e  into  the  plane-vector  |e', 
is  the  volume  of  the  parallelepiped  in  the  fig- 
ure, that  is,  TeTe'  sin  (angle  between  e  and  |e') 
=  TeTe'  cos  ^^.     Hence 

e|  e'  =  e'  I  e=l^m^-\-l^m^-\-l^m^=  TeTe  cos  f.  (106) 

If   Tez=  Te'  =  \,  l^  .  ,  .  l^,  m^  .  .  .  m^  are  di- 
rection cosines,  and  (105)  gives  a  proof  of  the 
formula  for  the  cosine  of   the  angle  between 
two  lines  in  terms  of  the  direction  cosines  of  the  lines.     We 
have  also  in  this  case 

ee'  =  {l,m,  -  l^m,)  \  i^  -f  {l^m,  -  l^m^  \  i^  +  {l^m,  -  l,m^  \  i^,  and, 
taking  the  co-square, 

(eej=  (sin  :')»=(/,/«,-  /,w,y +(/,///.-/,/;/,)' +(4w-A^0^  (^07) 
If  e|e'  =  o,  (108) 

e  is  parallel  to  the  plane-vector  perpendicular  to  e\  that  is,  e 
is  perpendicular  to  e' ,  as  is  also  shown  by  (106). 

Let  ;;  =  \e,r}'—  \e'  \  then 

;;|  7;'  =  1  e  .  e'  =  e'  I  e  =  e|  e'  =  TeTe'  cos  j'  =  Tr^Tr^'  q.qs  ;;'.  (109) 

and  7;|V  =  0  (no) 

is  the  condition  of  perpendicularity  of  two  plane-vectors.     Also 

either 

e|7;'  =  0,     or     r)'\e  =  0,  (ill) 

is  the  condition  that  a  vector  shall  be  perpendicular  to  a  plane- 
vector,  for  the  first  means  that  e  is  parallel  to  a  vector  which  is 


*  Grassmann  (1862),  Art.  335. 


e.^J«  = 


v<n.,Wn^  = 


(113) 


418  GRASSMANn's  space  analysis.  [Chap.  VIII. 

perpendicular  to  rf ,  and  the  second  that  //'  is  parallel  to  a  plane- 
vector  which  is  perpendicular  to  e. 

Equations  (7i)-(77)  of  Art.  9  become  stereometric  vector 
formulae  if  e^,  e,,  etc.,  be  substituted  iov  p^, p^,  etc.,  and  7,,  77,, 
etc.,  for  Zj,  Zj,  etc.    For  instance,  (76)  gives  the  vector  formulas 

For  lack  of  space  no  treatment  of  the  complement  in  a 
point  system  in  solid  space  is  given. 

Exercise  17. — To  prove  the   formulas  of  spherical  trigo- 
nometry cos  a  =  cos  d  cos  ^  +  sin  ^  sin  c  cos  Ay  and 
sin  a  _  sin  3  _  sin  c 
sin  ^  ~  sin  ^  ~  sin  C' 
Take  three  unit  vectors  e, ,  e,,  e,  parallel  to  the  radii  to  the 
vertices  of  the  spherical  triangle,  then  «=(angle  bet.  e,  and  €3), 
^=.(angle  bet.  e^e,  and  6^63),  etc.    In  eq.  (112)  put  €,€3  for  e/e/; 

hence    e^e,  |  e^eg  =  sin  d  sin  c  cos  A  =  e-.  e,  |  €3  —  ej  e^ .  ej  e^ 

=  cos  a  —  cos  d  cos  c. 
Again, 

or     sin  b  sin  c  sin  A  =  sin  a  sin  r  sin  B  =  sin  a  sin  /^  sin  C, 
whence  we  have  the  second  result  by  dividing  by  sin  a  sin  d  sin  c. 

Exercise   18. — Show  that  in  a  spherical  triangle  taken  as 

m  Exercise   17,  cos  -  = — ^=^ -t—ft r^>  whence  derive 

,  ,.  ,  /sins  sin  (^  —  a) 

the  ordinary  value  a  / — ^ \ 

Y         sin  o  sin  c 

Expanding,  the  numerator  becomes  i  +  Ue^eJ^  Ue^e^,  and 
the  denominator  1^2(1  +  Ue^e^\  Ue^e^).     Also  there  is  obtained 

Ue^eAUe^e^  =  ^=r^-^^^^-.     The  remainder  is  left  to  the  stu- 

dent. 

Prob.  24.  If  e^ ,  e^ ,  €3 ,  drawn  outward  from  a  point,  are  taken 
as  three  edges  of  a  tetrahedron,  show  that  the  six  planes  perpen- 


Art.  12.]  ADDITION    OF    SECTS   IN    SOLID    SPACE.  419 

dicular  to  the  edges  at  their  middle  points  all  pass  through  the  end 

of  the  vector  p  =  -^—ri  I  e^e,  .  e,-+  1  e^e^  .  e/  +  |  e,e, .  e,^).     (Sug- 

gestion.  We  must  have  (p  —  ^ej  |  e,  =  o,  with  two  other  similar 
expressions.) 

Prob.  25.  Show  that  e,  |  ee'  and  ee'.  |  e  are  three  mutually  per- 
pendicular vectors,  no  matter  what  the  directions  of  €  and  e' 
may  be. 

Prob.  26.  Let  e,,  e,,  e,  be  taken  as  in  Prob.  24  ;  let  A^  be  the 
area  of  the  face  of  the  tetrahedron  formed  by  joining  the  ends  of 
these  vectors,  and  2A^  =  Te^e^,  etc.;  also  6^j  =  Angle  between  e^e, 
and  €^€3,  etc.:  then  show  that  we  have  the  relation,  analogous  to 
that  of  Prob.  15,  Art.  8, 

'AJ'=  A,^-\-A^-\-  A^—  2A^A^  cos  ^,—  2A^A^  cos  S^—  2A^A^  cos  0^. 
If  6^1  ...  ^3  are  right  angles,  this  becomes  the  space-analog  of  the 
proposition  regarding  the  hypotenuse  and  sides  of  a  right-angled 
triangle.     (Suggestion.  2A,  =  T{e^  -  e^){€^  -  e,).) 

Prob.  27.  There  are  given  three  non-coplanar  lines  e^^e^,  ^^e, , 
^063;  planes  cut  these  lines  at  right  angles,  the  sum  of  the  squares  of 
their  distances  from  e^  being  constant.  Show  that  the  locus  of  the 
common  point  of  these  three  planes  is  (pI  eJ'+CPl  ^a)''+(Pl  ^3)'==<^'> 
if  re,  =  re,  =  re,  =  I. 

Art.  12.    Addition  of  Sects  in  Solid  Space. 

Two  lines  in  solid  space  vi^ill  not  in  general  intersect,  so  that 
their  sum  will  not  be,  as  in  eq.  (43),  a  definite  line.  For  let 
p^e^  and/,e,  be  any  two  sects:  then 

Ae,  +  Ae,  =  />xe,  +Ae,  +  e^e,  +  e,)  -  e,{e,  +  e,) 
=  ^o(e,  +  e,)  +  {p,  -  e,)e,  +  (/,  -  e,)e,; 
that  is,  the  sum  is  a  sect  passing  through  an  arbitrary  point  e^, 
and  a  plane-vector,  the  sum  of  the  two  in  the  equation.  The 
sum  cannot  be  a  single  sect  unless  the  two  are  coplanar ;  for  let 
/>,=/!  +  ;t^ei  +  ^e,  +  ^^^f  ^3  being  a  vector  not  parallel  to  Cje, ; 
hence      p.e^  +  p,e^  =  p^e^  +  (/,  +  xe^  +^e,  +  ^e3)e3 

=  /.  (e,  +  e,)  +  xe^  (e,  +  ej  +  2'e3e, 
=  [p.  +  xe^  (e,  +  ej  +  ^e3e, ; 
and  this  cannot  reduce  to  a  single  sect  unless  z  =  0,  that  is,  un- 
less/^e^  and/^e,  are  coplanar.     Since  a  plane-vector  is  a  line  at 


420  grassmann's  space  analysis.  [Chap.  VIIL 

00 ,  the  sum  of  two  lines  may  always  be  presented  as  the  sunr 
of  a  finite  line  and  a  line  at  oo . 

If  the  sum  of  any  two  sects  is  equal  to  the  sum  of  any 
other  two,  their  products  will  also  be  equal,  that  is,  the  two- 
pairs  will  determine  tetrahedra  of  equal  volumes.  For  let 
L^-\-  L^  —  L^-\-  L^\  then  squaring  we  have  L^L^  =  L^L^ ,  since 
ZjZj  =^  o,  etc. 

An  infinite  number  of  pairs  of  sects  can  be  found  such  that 
the  sum  of  each  pair  is  equal  to  the  sum  of  any  given  pair;  for 
let  a  given  pair  be/'^e,  -\- p^e^,  and  take  a  new  pair 

This  will  be  equal  to  the  given  pair  if  we  have 
x,u^+y,v^  =  ^^'^1+y^'^.,  =  i>  and  x,u^  +y,v,  =  x^u^  +y,'^^  =  o). 

Since  there  are  eight  arbitrary  quantities  with  only  four 
equations  of  condition,  the  desired  result  can  evidently  be  ac- 
complished in  an  infinite  number  of  ways. 

Let  p^€^ ,  p^e^  ....  p„6„  be  n  sects,  and  let  5  be  their  sum,, 
and  e^  any  point,  then 

S=2pe  =  e^2e  -  e,2e  +  ^pe  =  e,^e  +  ^(/  -  e,)e,  ....(113) 

1 
the  sum  of  a  sect  and  a  plane-vector  as  before. 

If  2{p  —  e^e  is  parallel  to  ^'e  it  may  be  written  as  the  prod- 
uct of  some  vector  e'  into  ^e,  that  is,  e'^e,  when  the  sum  be- 
comes 5  =  e^'Se  +  e'^e  =  (e'  +  e')^€,  a  sect,  because  e^  -\-  e  is 
a  point.  In  no  other  case  does  5  reduce  to  a  single  sect.  If 
^e  =  o,  5  becomes  a  plane-vector.  Of  the  two  parts  compos- 
ing 5,  tlie  sect  will  be  unchanged  in  magnitude  and  direction  if 
e^  be  moved  to  a  new  position,  while  the  plane-vector  will  in 
general  be  altered.  It  is  proposed  to  show  that  a  point  q  may 
be  substituted  for  e^  such  that  the  plane-vector  will  be  perpen- 
dicular to  ^e.     Writing 

S^q^e-{q-  ^J^e  +  ^{p  -  ^e, 
and,  for  brevity,  putting  q  —  e^  =  p,  :2e  =  a,  :2{p  —  e^e  =  \ftr 

so  that 

S^qa  —  f>a  +  \^,  (ll4> 


Art.  13.]  addition  of  sects  in  solid  space,  421 

we  must  have  for  perpendicularity,  by  (in), 

(I yS  —  pa)  I  or  =  O  =  I  fta  —  pa.\a,  — 

or  pa  .\a  =  a.p\a  —  p,a-  =  \/3a,  (nS) 

The  second  member  is  obtained  from  the  first  by  substitut- 
ing in  eq.  (74)  p  for/>,  and  a  for/, and  ^,,  in  accordance  with  the 
statement  at  the  end  of  Art.  11.  If  in  (11 5)  we  make  p\a  =  o, 
p  will  be  the  vector  from  e^  to  ^  taken  perpendicularly  to  a, 
say 

A  =  k^  -^  «-=  ^,  -  ^0.  (116) 

Since  a  and  /3  are  known,  the  required  point  has  been 
found.     Multiply  (115)  by  a;  then,  using  (75), 

—  ap .  a-  =  pa .  a-  =  a\ /3a  =^  \ /3 ,  a-  —  \a,  a\/3, 
whence,  substituting  in  (114), 

This  may  be  called  the  normal  form  of  S* 

The  sects  of  this  article  represent  completely  the  geometric 
properties  of  forces,  hence  all  that  has  been  shown  applies 
immediately  to  a  system  of  forces  in  solid  space.  We  have 
only  to  substitute  the  words  force  and  couple  for  sect  and  plane- 
vector.  The  resultant  action  of  any  system  of  forces  is  5, 
called  by  Ball  in  his  Theory  of  Screws  "  a  wrench."  The  con- 
dition for  equilibrium  is  S  =  o,  which  gives  at  once 

2e  =  o     and     ^(p  —  e^)e  =  o;  (118) 

since  otherwise  we  must  have  e^^e  =  —  2{/>  —  ^Je,  which  is 
an  impossibility.  The  line  g^e  is  the  central  axis  of  the  sys- 
tem of  forces  5. 

Lack  of  space  forbids  a  further  development  of  the  subject, 
but  what  has  been  given  in  this  article  will  indicate  the  perfect 
adaptability  of  this  method  to  the  requirements  of  mechanics. 

Exercise  19. — Reduce  /,e,  -|-  p^e^  =  5  to  its  normal  form. 
•S"  =  ^o(e,  +  O  +  (A  -  Oe,  +  (A  -  ^oK'  Foi*  convenience 
suppose  A  ^^d  A  to  be  taken  at  the  ends  of  the  common  per- 

*Grassmann  (1862),  Art.  346. 


428  grassmann's  space  analysis.  [Chap.  VIIL 

pendicular  on/^e,  and/^e^,  and  moreover  let  e^  =  J(/^  +A)» 
p^  —  e^  —  I  z=z  —  [p^  —  ^ J  ;  then  z|e,  =  z|e,  =  O.     Accordingly 

5  ^  ..(e.  +  e,)  +<e.  -  e.)  =  ?(6.+6j+  ^''|,'ff;)7''^  ■ '  6.+6,> 

t.(e.-e.)|(e.+ej  _  e.^-e/ 

Hence  the  normal  form  of  5  is 

Exercise  20. — Forces  are  represented  by  the  six  edges  of  a 
tetrahedron  e^e^,  e^e^,  e^e^,  e^e^,  e^e^,  e^e^\  find  the  5,  reduce  to 
normal  form,  and  consider  the  special  case  when  three  diedral 
angles  are  right  angles.  5  =  e,{e^  +  ^,  +  e^  +  e^e^  +  e^e^  +  e,e^ 
=  ^o(e,+e,+e,)+(^,-^,)(^,-^J=  ^„(e,+e,+e3)  +  (e,-e,)(e3-e,) 
=  ^o(e,  +  e,  +  €3)  +  e,e,  +  e,e,  +  e,e, ,  in  which  e,  =  ^,  -  i?„ , 
etc.     Hence 

For  the  rectangular  tetrahedron  let  e,  =  ai^ ,  e,  =  ^z, , 
€3  ==  ci^ »  ^,  >  ^2,  ^9  being  unit  normal  vectors.     Then  we  find 

Exercise  21. — A  pole  50  feet  high  stands  on  the  ground  and 
is  held  erect  by  three  guy-ropes  symmetrically  arranged  about 
it,  attached  to  its  top  and  to  pegs  in  the  ground  50  feet  from 
the  pole.  The  wind  blows  against  the  pole  with  a  pressure  of 
50  pounds  in  the  direction  e^  —  p,  when  e^  is  at  the  bottom  of 


Art.  12.]  addition  of  sects  in  solid  space.  423 

the  pole,  and  /  divides  the  distance  between  two  of  the  pegs 

in  the  ratio  — :  find  the  tension  on  the  guys  and  the  pressure 

on  the  ground. 

Evidently  only  two  of  the  guys  will  be  in  tension  ;  let  their 
pegs  be  at  e^  and  e^,  and  let  e^  be  at  the  top  of  the  pole,  and  w 

7fl€    — f—  Tl€ 

the  weight  of  the  pole.     Then/  ==  — ^—7- — ^,  and  the  equation 
of  equilibrium  is 

Te,e,  =  50,  Te^e,  =  Te,e,  =  50  V2,  T{p  -  e,)  =  T^f-^Jf^'  "^o) 

50 


and  e,  =  [/(e,  —  e,);  then  T(p  —  e,)  =        ,       Vm''  +  «"  —  mn, 

because  e,-  =  e-,^  =  i,  and  e^\e^  =  cos  120°  =  —  J.    Hence  the 
equation  of  equilibrium  becomes 

\m  -\-  n  —  mn  V2  V2 

Multiply  successively  by  ^,^,,  e^e^y  and  e^e^^  and  we  obtain 

x-\-w_y_2_  25 

m-\-  n      m  V2      n  V2       Vm^  -\-  n^  —  mn 

y  and  z  being  the  tensions,  and  x  -\-  w  the  upward  pressure. 

Prob.  28.  Three  equal  poles  are  set  up  so  as  to  form  a  tripod, 
and  are  mutually  perpendicular;  a  weight  w  hangs  upon  a  rope 
which  passes  over  a  pulley  at  the  top  of  the  tripod,  and  thence 

down  under  a  pulley  at  the  ground  at  a  point/ =  ^/^,  in  which 
^, .  .  .  ^3  are  at  the  feet  of  the  poles,  and  ^/  =  i ;  if  the  rope  is  pulled 


424  grassmann's  space  analysis.  [Chap.  VIII. 

so  as  to  raise  a/,  show  that  the  pressures  on  the  poles,  supposing  the 
pulleys  frictionless,  are 

^(A+4^)'  K^"+^)'  '"['h^^vi)- 

Prob.  29.  Six  equal  forces  act  along  six  successive  edges  of  a 
cube  which  do  not  meet  a  given  diagonal;  show  that  if  the  edges  of 
the  cube  be  parallel  to  z^,  z^,  Zg,  and  F  be  the  magnitude  of  each 
force,  then  S=  —  2F\  {i^  +  ^3  +  ^3),  if  the  diagonal  taken  be  parallel 
to  z,  +  z,  +  ^,. 

Prob.  30.  Three  forces  whose  magnitudes  are   i,  2,  and  3  act 
along  three  successive  non-coplanar  edges  of  a  cube;  show  that  the 
normal  form  of  S  is 
S=(.e,+  \it,  +  K  -  ^t,){i,  +  21,+  31.)+  Al  (',  +  2'.  +  3^)- 

Prob.  31.  Forces  act  at  the  centroids  of  the  faces  of  a  tetrahedron, 
perpendicular  and  proportional  to  the  faces  on  which  they  act,  and 
all  directed  inwards,  or  else  all  outwards;  show  that  they  are  in 
equilibrium. 


Art.  l.J  INTRODUCTION,  425 


Chapter  IX. 
VECTOR  ANALYSIS   AND    QUATERNIONS. 

By  Alexander  Macfarlane, 
Lecturer  in  Electrical  Engineering  in  Lehigh  University. 

Art.  1.    Introduction. 

By  "  Vector  Analysis  "  is  meant  a  space  analysis  in  which 
the  vector  is  the  fundamental  idea;  by ''Quaternions"  is  meant 
a  space-analysis  in  which  the  quaternion  is  the  fundamental 
idea.  They  are  in  truth  complementary  parts  of  one  whole; 
and  in  this  chapter  they  will  be  treated  as  such,  and  developed 
so  as  to  harmonize  with  one  another  and  with  the  Cartesian 
Analysis.*  The  subject  to  be  treated  is  the  analysis  of  quanti- 
ties in  space,  whether  they  are  vector  in  nature,  or  quaternion 
in  nature,  or  of  a  still  different  nature,  or  are  of  such  a  kind  that 
they  can  be  adequately  represented  by  space  quantities. 

Every  proposition  about  quantities  in  space  ought  to  re- 
main true  when  restricted  to  a  plane ;  just  as  propositions 
about  quantities  in  a  plane  remain  true  when  restricted  to  a 
straight  line.  Hence  in  the  following  articles  the  ascent  to  the 
algebra  of  space  is  made  through  the  intermediate  algebra  of 
the  plane.  Arts.  2-4  treat  of  the  more  restricted  analysis, 
while  Arts.  5-10  treat  of  the  general  analysis. 

This  space  analysis  is  a  universal  Cartesian  analysis,  in  the 
same  manner  as  algebra  is  a  universal  arithmetic.  By  provid- 
ing an  explicit  notation  for  directed  quantities,  it  enables  their 
general  properties  to  be  investigated  independently  of  any 
particular  system  of  coordinates,  whether  rectangular,  cylin- 
drical, or  polar.     It  also  has  this  advantage  that  it  can  express 

*For  a  discussion  of  the  relation  of  Vector  Analysis  to  Quaternions,   sse 
Nature,  1891-1893. 


42G  VECTOR    ANALYSIS   AND   QUATERNIONS.  [Chap.  IX. 

the  directed  quantity  by  a  linear  function  of  the  coordinates^ 
instead  of  in  a  roundabout  way  by  means  of  a  quadratic  func- 
tion. 

The  different  views  of  this  extension  of  analysis  which  have 
been  held  by  independent  writers  are  briefly  indicated  by  the 
titles  of  their  works : 

Argand,  Essai  sur  une  maniere  de  representer  les  quantites 
imaginaires  dans  les  constructions  geometriques,  1806. 

Warren,  Treatise  on  the  geometrical  representation  of  the  square 
roots  of  negative  quantities,  1828. 

Moebius,  Der  barycentrische  Calcul,  1827. 

Bellavitis,  Calcolo  delle  Equipollenze,  1835. 

Grassmann,  Die  lineale  Ausdehnungslehre,  1844. 

De  Morgan,  Trigonometry  and  Double  Algebra,  1849. 

O'Brien,  Symbolic  Forms  derived  from  the  conception  of  the 
translation  of  a  directed  magnitude.  Philosophical  Transactions, 
1851. 

Hamilton,  Lectures  on  Quaternions,  1853,  and  Elements  of 
Quaternions,  1866. 

Tait,  Elementary  Treatise  on  Quaternions,  1867. 

Hankel,  Vorlesungen  iiber  die  complexen  Zahlen  und  ihre 
Functionen,  1867. 

Schlegel,  System  der  Raumlehre,  1872. 

Hoiiel,  Theorie  des  quantites  complexes,  1874. 

Gibbs,  Elements  of  Vector  Analysis,  1881-4. 

Peano,  Calcolo  geometrico,  1888. 

Hyde,  The  Directional  Calculus,  1890. 

Heaviside,  Vector  Analysis,  in  "  Reprint  of  Electrical  Papers," 
1885-92. 

Macfarlane,  Principles  of  the  Algebra  of  Physics,  1891.  Papers 
on  Space  Analysis,  189 1-3. 

An  excellent  synopsis  is  given  by  Hagen  in  the  second  volume 
of  his  "  Synopsis  der  hoheren  Mathematik." 

Art.  2.    Addition  of  Coplanar  Vectors. 

By  a  "  vector  "  is  meant  a  quantity  which  has  magnitude 
and  direction.     It  is  graphically  represented  by  a  line  whose 


Art.  2.]  addition  of  coplanar  vectors.  427 

length  represents  the  magnitude  on  some  convenient  scale,  and 
whose  direction  coincides  v^rith  or  represents  the  direction  of 
the  vector.  Though  a  vector  is  represented  by  a  line,  its 
physical  dimensions  maybe  different  from  that  of  a  line.  Ex- 
amples are  a  linear  velocity  which  is  of  one  dimension  in 
length,  a  directed  area  which  is  of  two  dimensions  in  length, 
an  axis  which  is  of  no  dimensions  in  length. 

A  vector  will  be  denoted  by  a  capital  italic  letter,  as  ^,*  its 
magnitude  by  a  small  italic  letter,  as  b,  and  its  direction  by  a  small 
Greek  letter,  as  y5.  For  example,  B  —  b^,  R=  rp.  Sometimes 
it  is  necessary  to  introduce  a  dot  or  a  mark  /  to  separate 
the  specification  of  the  direction  from  the  expression  for  the 
magnitude  ;f  but  in  such  simple  expressions  as  the  above,  the 
difference  is  sufficiently  indicated  by  the  difference  of  type.  A 
system  of  three  mutually  rectangular  axes  will  be  indicated, 
as  usual,  by  the  letters  i,j\  k. 

The  analysis  of  a  vector  here  supposed  is  that  into  magni- 
tude and  direction.  According  to  Hamilton  and  Tait  and 
other  writers  on  Quaternions,  the  vector  is  analyzed  into  tensor 
and  unit-vector,  which  means  that  the  tensor  is  a  mere  ratio 
destitute  of  dimensions,  while  the  unit-vector  is  the  physical 
magnitude.  But  it  will  be  found  that  the  analysis  into  magni- 
tude and  direction  is  much  more  in  accord  with  physical  ideas, 
and  explains  readily  many  things  which  are  difficult  to  explain 
by  the  other  analysis. 

A  vector  quantity  may  be  such  that  its  components  have  a 
common  point  of  application  and  are  applied  simultaneously; 
or  it  may  be  such  that  its  components  are  applied  in  succes- 
sion, each  component  starting  from  the  end  of  its  predecessor. 
An  example  of  the  former  is  found  in  two  forces  applied  simul- 
taneously at  the  same  point,  and  an  example  of  the  latter  in 

*  This  notation  is  found  convenient  by  electrical  writers  in  order  to  harmo- 
nize with  the  Hospitalier  system  of  symbols  and  abbreviations. 

fThe  dot  was  used  for  this  purpose  in  the  author's  Note  on  Plane  Algebra^ 
1883;  Kennelly  has  since  used  Z  for  the  same  purpose  in  his  electrical  papers 


428  VECTOR   ANALYSIS   AND   QUATERNIONS.  [Chap.  IX. 

two  rectilinear  displacements  made  in  succession  to  one  an- 
other. 

Composition  of  Components  having  a  common  Point  of 
Application. — Let  OA  and  OB  represent  two  vectors  of  the 
same  kind  simultaneously  applied  at  the  point  O.     Draw  BC 
Q  parallel  to  OA,  and  AC   parallel  to  OB,  and 
join  OC.     The  diagonal  OC  represents  in  mag- 
nitude and  direction  and  point  of  application 
o"  "A      the  resultant  of  OA  and  OB.     This  principle 

was  discovered  with  reference  to  force,  but  it  applies   to  any 
vector  quantity  coming  under  the  above  conditions. 

Take  the  direction  of  OA  for  the  initial  direction  ;  the  di- 
rection of  any  other  vector  will  be  sufficiently  denoted  by  the 
angle  round  which  the  initial  direction  has  to  be  turned  in 
order  to  coincide  with  it.  Thus  OA  may  be  denoted  by 
fJ_o,  OB  by/,/^,,  OC  by//^.  From  the  geometry  of  the  fig- 
ure it  follows  that 

/'=/■'+/,' +  2/,/,  cos  ^, 

/,  Sin  S,      . 


and  tan  H 


/.  +/,  cos  s; 


Jience  OC  =  -//.'  +  /,'  +  2/,/,  cos  d^  /^^jA^^. 

Example. — Let  the  forces  applied  at  a  point  be  2/0"  and 
3/60°.      Then   the  resultant  is  i/4  +  9+12  X  i  /tan"^  ?-— ^ 

=  4»  36/36°  30^ 

If  the  first  component  is  given  as/, /6/,,  then  we  have  the 
more  symmetrical    formula 
OC  =  V/."+/,'  + 2/./,  COS  (..-.,  /tan:>^i^A±A|L|., 

When  the  components  are  equal,  the  direction  of  the  re- 
•sultant  bisects  the  angle  formed  by  the  vectors ;  and  the  mag- 
nitude of  the  resultant  is  twice  the  projection  of  either  compo- 
nent on  the  bisecting  line.     The  above  formula  reduces  to 

OC  =  2/  cos  ^  /-\ 
■^'         2/  2 


Art.  2.]  addition  of  coplanar  vectors,  429 

Example. — The  resultant  of  two  equal  alternating  electro- 
motive forces  which  differ  120°  in  phase  is  equal  in  magnitude 
to  either  and  has  a  phase  of  60°. 

Given  a  vector  and  one  component,  to  find  the  other  com- 
ponent.— Let  OC  represent  the  resultant,  and  OA  the  compo- 
nent. Join  AC  and  draw  OB  equal  and  g  ^ 
parallel  to  AC.  The  line  OB  represents 
the  component  required,  for  it  is  the  only  y^' 
line  which  combined  with  OA  gives  OC  ^  o  ^ 
as  resultant.  The  line  OB  is  identical  with  the  diagonal  of  the 
parallelogram  formed  by  OC  and  OA  reversed  ;  hence  the  rule 
is,  *'  Reverse  the  direction  of  the  component,  then  compound 
it  with  the  given  resultant  to  find  the  required  component." 
Let  f/Q  be  the  vector  and  fjo  one  component ;  then  the 
other  component  is 

m = v/'+/'  -  ^//.  cos  e/urri_/^yl^ ,. 

Given  the  resultant  and  the  directions  of  the  two  compo- 
nents, to  find  the  magnitude  of  the  components. — The  resultant 
is  represented  by  OC,  and  the  directions  by  OX  and  OY. 
Y,  From   C   draw  CA  parallel  to  OY,  and  CB 

parallel  to  OX ;  the  lines  OA  and  OB  cut 
off  represent  the  required  components.  It 
is  evident  that  OA  and  OB  when  com- 
pounded  produce  the  given  resultant  OC, 
and  there  is  only  one  set  of  two  components  which  produces 
a  given  resultant ;  hence  they  are  the  only  pair  of  components 
having  the  given  directions. 

het//6  be  the  vector  and  /d^  and  /d^  the  given  directions^ 

Then 

/,  +/,  cos  {H,  -  0,)  =/cos  (0  -  e,), 

/,  COS  (6,  -  e,)  +/,  =/cos  (e,  -  ff), 

from  which  it  follows  that 

jcos  (6  -  e,)  -  cos  (e,  -  6)  cos  (8,  —  d,)  } 
^'~^  I  -  cos'  (^y,  -  d,) 


430  VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChAP.  IX. 

For  example,  let  100/60°,  /30°,  and  790°  be  given ;  then 

cos  30° 

f.—  100 — i .  o> 

•^  ^  I  +  cos  60 

Composition  of  any  Number  of  Vectors  applied  at  a  com- 
mon Point. — The  resultant  may  be  found  by  the  following 
graphic  construction  :  Take  the  vectors  in  any  order,  sls  A,B,  C. 
From  the  end  of  A  draw  B'  equal  and  par- 
allel to  By  and  from  the  end  of  B^  draw  C^ 
.g  equal  and  parallel  to  C;  the  vector  from 
the  beginning  of  A  to  the  end  of  C  is  the 
resultant  of  the  given  vectors.  This  follows 
"a i  by  continued  application  of  the  parallelo- 
gram construction.  The  resultant  obtained  is  the  same,  what- 
ever the  order;  and  as  the  order  is  arbitrary,  the  area  enclosed 
has  no  physical  meaning. 

The  result  may  be  obtained  analytically  as  follows : 
Given  /,/e,  +/y^,  +73/^3  +  .  .  .  +  /n  /^n. 

Now  /,/e,  =/,cos  ^,/o+/,sin  6, ^~. 

Similarly  /,/^  =  /,  cos  e,/o_-\-/,  sin  d^  /^, 

and  /n/6^  =  /„  cos  0„/o  ^/„  sin  0„  /^. 

Hence  2\//e]  =  JJS/cos  6]  /o-\-  {2fsm  6}    /- 

=  V(2fcos  0\^  (2/ sin  Oy  ,  tan-^^^^. 

In  the  case  of  a  sum  of  simultaneous  vectors  applied  at  a  com- 
mon point,  the  ordinary  rule  about  the  transposition  of  a  term  in 
an  equation  holds  good.  For  example,  if  A  -{-B  -}-  C  =  o,then 
A  -{-  B  =^  -^  Cy  and  A  -^  C  =  -  B,  and  B  -{-  C  =  -  A,  etc. 
This  is  permissible  because  there  is  no  real  order  of  succession 
among  the  given  components.* 

*  This  does  not  hold  true  of  a  sum  of  vectors  having  a  real  order  of  succes- 
sion.    It  is  a  mistake  to  attempt  to  found  space-analysis  upon  arbitrary  formal 


Art.  2.]  ADDITION    OF    COPLANAR    VECTORS.  431 

Composition  of  Successive  Vectors. — The  composition  of 
successive  vectors  partakes  more  of  the  nature  of  multiplica- 
tion than  of  addition.  Let  ^  be  a  vector  start- 
ing from  the  point  O,  and  B  a  vector  starting 
from  the  end  of  A,  Draw  the  third  side  OP,  / 
and  from  O  draw  a  vector  equal  to  B^  and  from 
its  extremity  a  vector  equal  to  A.  The  line  OP  is  not  the 
complete  equivalent  of  ^  +  ^;  if  it  were  so,  it  would  also  be 
the  complete  equivalent  of  B  -\-  A.  But  A  -{-  B  and  B  -{-A 
determine  different  paths ;  and  as  they  go  oppositely  around, 
the  areas  they  determine  with  OP  have  different  signs.  The 
diagonal  OP  represents  A  -\-  B  only  so  far  as  it  is  consid- 
ered independent  of  path.  For  any  number  of  successive 
vectors,  the  sum  so  far  as  it  is  independent  of 
path  is  the  vector  from  the  initial  point  of  the 
first  to  the  final  point  of  the  last.  This  is  also 
true  when  the  successive  vectors  become  so  small 
as  to  form  a  continuous  curve.  The  area  between 
the  curve  OPQ  and  the  vector  OQ  depends  on  the  path,  and 
has  a  physical  meaning. 

Prob.  I.  The  resultant  vector  is  123/45°,  and  one  component 
is  100/0°;  find  the  other  component. 

Prob.  2.  The  velocity  of  a  body  in  agiven  plane  is  200  /75°,  and 
one  component  is  100/25°;  find  the  other  component. 

Prob.  3.  Three  alternating  magnetomotive  forces  are  of  equal 
virtual  value,  but  each  pair  differs  in  phase  by  120°;  find  the  re- 
sultant.    (Ans.  Zero.) 

Prob.  4.  Find  the  components  of  the  vector  100/70°  in  the  direc- 
tions 20°  and  100°. 

Prob.  5.  Calculate  the  resultant  vector  of  1/10°,  2/20°,  3/30°, 
4/40°' 

Prob.  6.  Compound  the  following  magnetic  fluxes:  k  sin  «/  -J- 
^sin  («/  —  i2o°)/i2o°  +  ^  sin  (nf  —  24o°)/24o°.     (Ans.  \h/nt.) 

laws;  the  fundamental  rules  must  be  made  to  express  universal  properties  of  the 
thing  denoted.  In  this  chapter  no  attempt  is  made  to  apply  formal  laws  to 
directed  quantities.     What  is  attempted  is  an  analysis  of  these  quantities. 


433  VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChaP.  IX. 

Prob.  7.  Compound  two  alternating  magnetic  fluxes  at  a  point, 

a  cos  nt  /o  and  a  sin  nt  /  — .      (Ans.  a  /ni.) 

/    2 

Prob  8.  Find  the  resultant  of  two  simple  alternating  electromo- 
tive forces  100/20°  and  50/75°. 

Prob.  9.  Prove  that  a  uniform  circular  motion  is  obtained  by- 
compounding  two  equal  simple  harmonic  motions  which  have  the 
space-phase  of  their  angular  positions  equal  to  the  supplement  of  the 
time-phase  of  their  motions. 

Art.  3.  Products  of  Coplanar  Vectors. 

When  all  the  vectors  considered  are  confined  to  a  common 
plane,  each  may  be  expressed  asthe  sum  of  two  rectangular 
components.  Let  i  and/  denote  two  directions  in  the  plane  at 
right  angles  to  one  another  ;  then  A  =  a  J-  -\-  a^j\  B  =  dj  +  Kjy 
R  —  xiA^yj.  Here  i  and  7  are  not  unit-vectors,  but  rather 
signs  of  direction. 

Product  of  two  Vectors.  — Let  A=^  <^i^+^ay  and  B  =  b^-\-bJ 
be  any  two  vectors,  not  necessarily  of  the  same  kind  physically. 
We  assume  that  their  product  is  obtained  by  applying  the 
distributive  law,  but  we  do  not  assume  that  the  order  of  the 
factors  is  indifferent.  Hence 
AB  =  {a J  +  aJ){bJ,-\-  bj)  =  ajb^ii  +  aJbJj-\-  ajj^ij  -\-aJ)Ji, 

If  we  assume,  as  suggested  by  ordinary  algebra,  that  the 
square  of  a  sign  of  direction  is  -|->  and  further  that  the  product 
of  two  directions  at  right  angles  to  one  another  is  the  direction 
normal  to  both,  then  the  above  reduces  to 

AB  =  a  J),  +  aj}^  +  {afi^  —  a,b^)k. 

Thus  the  complete  product  breaks  up  into  two  partial 
products,  namely,  a^b^-  -\-  aj?^  which  is  independent  of  direc- 
tion, and  {a^b^  —  aj?^k  which  has  the  axis  of  the  plane  for 
direction.* 

*  A  common  explanation  which  is  given  of  ij  =  k  \s  that  i  is  an  operator,/ an 
operand,  and  k  the  result.  The  kind  of  operator  which  i  is  supposed  to  denote 
is  a  quadrant  of  turning  round  the  axis  i  ;  it  is  supposed  not  to  be  an  axis,  but 
a  quadrant  of  rotation  round  an  axis.  This  explains  the  result  iJ  ■=  k,  but 
unfortunately  it  does  not  explain  ii  =  +  ;  for  it  would  give  ii  =  i. 


Art.  3.]  PRODUCTS  OF  coplanar  vectors.  43S 

Scalar  Product  of  two  Vectors. — By  a  scalar  quantity  is 
meant  a  quantity  which  has  magnitude  and  may  be  positive  or 
negative  but  is  destitute  of  direction.  The  former  partial 
product  is  so  called  because  it  is  of  such  a  nature.  It  is 
denoted  by  SAB  where  the  symbol  S,  being  in  Roman  type^ 
denotes,  not  a  vector,  but  a  function  of  the 
vectors  A  and  B.  The  .geometrical  mean- 
ing of  SAB  is  the  product  of  A  and  the 
orthogonal  projection  of  B  upon  A.  Let 
OP  and  OQ  represent  the  vectors  A  and  B; 
draw    QM  and  NL  perpendicular  to   OP. 

Then 

(OP)(OM)  =  (OP)(OL)  +  (OP)(LM), 


-^.^■+*»^l' 


Corollary  i. — SEA  =  S^^.  For  instance,  let  A  denote  a 
force  and  B  the  velocity  of  its  point  of  application  ;  then  S^-^ 
denotes  the  rate  of  working  of  the  force.  The  result  is  the 
same  whether  the  force  is  projected  on  the  velocity  or  the 
velocity  on  the  force. 

Example  I. — A  force  of  2  pounds  East  +  3  pounds  North  is 
moved  with  a  velocity  of  4  feet  East  per  second  +  5  feet  North 
per  second ;  find  the  rate  at  which  work  is  done. 

2X4+3X5=  23  foot-pounds  per  second.  > 

Corollary  2. — A^  =t  a^  -\-  a^  —  a".  The  square  of  any  vector 
is  independent  of  direction ;  it  is  an  essentially  positive  or 
signless  quantity  ;  for  whatever  the  direction  of  A,  the  direction 
of  the  other  ^  must  be  the  same;  hence  the  scalar  product 
cannot  be  negative. 

Example  2. — A  stone  of  10  pounds  mass  is  moving  with  a 
velocity  64  feet  down  per  second  +  100  feet  horizontal  per 
second.    Its  kinetic  energy  then  is 

—  (64'  +  100')  foot-poundals, 


--Gr, -► 


434  VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChAP.  IX. 

a  quantity  which  has  no  direction.  The  kinetic  energy  due  to 

64' 
the  downward  velocity  is  10  X  —  and  that  due  to  the  hori- 
zontal velocity  is  —  X  100'* ;  the  whole  kinetic  energy  is   ob- 
tained, not  by  vector,  but  by  simple  addition,  when  the  com- 
ponents are  rectangular. 

Vector  Product  of  two  Vectors. — The  other  partial  product 
from  its  nature  is  called  the  vector  product,  and  is  denoted  by 
VAB.  Its  geometrical  meaning  is  the 
product  of  A  and  the  projection  of  i?  which 
is  perpendicular  to  Ay  that  is,  the  area  of 
the  parallelogram  formed  upon  A  and  B. 
Let  OP  and  OQ  represent  the  vectors  A 
"*  and  Bj  and  draw  the  lines  indicated  by  the 
figure.  It  is  then  evident  that  the  area 
of  the  triangle  OPQ  =  a,d,  —  ^a,a^  —  ^b^b^  —  j^{a,  —  b,){b,  —  a^\ 

Thus  {a^b^  —  a^b^)k  denotes  the  magnitude  of  the  parallelo- 
gram formed  by  A  and  B  and  also  the  axis  of  the  plane  in 
which  it  lies. 

It  follows  that  VBA  =  —  VAB.  It  is  to  be  observed 
that  the  coordinates  of  A  and  B  are  mere  component  vectors, 
whereas  A  and  B  themselves  are  taken  in  a  real  order. 

Example. — Let  A  =  (lo^  +  1 17)  inches  and  B  =  (5^  +  12/) 
inches,  then  VAB  =  (120  —  SS)^  square  inches;  that  is,  65 
square  inches  in  the  plane  which  has  the  direction  k  for  axis. 

If  A  is  expressed  as  aa  and  B  as  b^,  then  SAB  =  ab  cos  a/S, 
where  aj3  denotes  the  angle  between  the  directions  a  and  /3. 

Example. — The  effective  electromotive  force  of  100  volts 
per  inch  790°  along  a  conductor  8  inch  745°  is  S^^  =  8  X  lOO 
cos /45°  /9o°  volts,  that  is,  800  cos  45°  volts.  Here 745°  indicates 
the  direction  or  and  790°  the  direction  /3,  and  745°  79^°  means 
the  angle  between  the  direction  of  45°  and  the  direction  of  90°. 

Also  VAB=  ab  sin  ap  .  a^,  where  a^  denotes  the  direction 
which  is  normal  to  both  a  and  /?,  that  is,  their  pole. 


Art.  3.]  products  of  coplanar  vectors.  435 

Example. — At  a  distance  of  lo  feet  /30°  there  is  a  force  of 
lOO  pounds  /6o°.     The  moment  is  YAB 


=  lox  loo  sin  /30°  /6o°  pound-feet  90°/  790° 
=  1000  sin  30°  pound-feet  90°/  790°. 
Here  90°/  specifies  the  plane  of  the  angle  and  /90°  the  angle. 
The  two  together  written  as  above  specify  the  normal  k. 

Reciprocal  of  a  Vector. — By  the  reciprocal  of  a  vector  is 
meant  the  vector  which  combined  with  the  original  vector  pro- 
duces the  product  -j"  i«  The  reciprocal  of  A  is  denoted 
by  A~\  Since  AB  =  ab  (cos  a^-\-s\n  a^ .  ap),  b  must  equal 
a~^  and  p  must  be  identical  with  a  in  order  that  the  product 
may  be  i.     It  follows  that 

...  _\^  _ctoL  _al±aj_ 
a         a         a^   +  a^ 

The  reciprocal  and  opposite  vector  is  —  A~^,     In  the  figure 
let  OP  =  2^  be  the  given  vector ;  then  OQ  =  \p  is  its  recipro- 
cal, and  OR  =  ■J(  —  /?)  is  its  reciprocal  and 
opposite.''^  R         Q  P 

Example.—If  ^  =   10  feet  East  +  5   feet  North,  A'^  = 

—    feet    East  +  —   feet    North    and  —  ^-*  =  ^^^  feet 
125  '     125  125 

East  —   —  feet  North. 
125 

Product  of  the  reciprocal  of  a  vector  and  another  vector. — 

A-'B  =  \aB, 
a 

=  ^\^A  +  a  A  +  {a  A  -  a  A)  ^\y 

=  -  (cos  aft  +  sin  aft .  aft). 

*  Writers  who  identify  a  vector  with  a  quadrantal  versor  are  logically  led  to 
define  the  reciprocal  of  a  vector  as  being  opposite  in  direction  as  well  as  recip- 
rocal in  magnitude. 


436  VECTOR    ANALYSIS    AND    QUATERNIONS.  [Chap.  IX. 

b  b  

Hence  S^"*^  =  -cos  a^  and  NA-'B  =  -sin  a6 ,  a6. 

Product  of  three  Coplanar  Vectors. — Let  A  =  a  J  -f-  a^j^ 
B  =  bj  +  b^j,  C  =  cj  +  c^j  denote  any  three  vectors  in  a 
common  plane.     Then 

iAB)C  =  {{a,b,  +  aA)  +  (^.^.  -  ^AWi^.^  +  cj) 

The  former  partial  product  means  the  vector  C  multiplied 
R^  by  the  scalar  product  of  A  and  B  ;  while  the 

latter   partial   product   means  the  comple- 
te mentary  vector  of  C  multiplied  by  the  mag- 
^^^^^  .. ^  nitude  of  the  vector  product  of  A  and  B. 

io  1   I^  these  partial  products  (represented  by  OP 

and  OQ)  unite  to  form  a  total  product,  the  total  product  will  be 
represented  by  OR,  the  resultant  of  OP  and  OQ. 

The  former  product  is  also  expressed  by  S^^ .  C,  where  the 
point  separates  the  vectors  to  which  the  S  refers  ;  and  more 
analytically  by  abc  cos  a^ .  y. 

The  latter  product  is  also  expressed  by  {yAB)Cj  which  is 
equivalent    to    V{'VAB)Cy  because  YAB  is  at    right    angles 

to  C,  It  is  also  expressed  by  abc  sin  a/3,  apy,  where  a/3y  de- 
notes the  direction  which  is  perpendicular  to  the  perpendicular 
to  a  and  /3  and  y. 

If  the  product  is  formed  after  the  other  mode  of  association 
we  have 

A{BC)={a,t  +  aJ)(b,c,  +  b^c^)  +  {aj  +  aj){b,c,  -  b,c,)k 
=  (^1^1  +  ^.<^JK^'  +  a.j)  +  i^^c,  -  b^c^{aji  —  a,j) 
=^SBC.A  +VA(YBC). 

The  vector  aj  —  a  J  is  the  opposite  of  the  complementary 
vector  of  aj  +  a^j.  Hence  the  latter  partial  product  differs 
with  the  mode  of  association. 

Example.— Let  A    =  \/o_-\-  2/90°,     B  =  3/^°+  4/90°, 

C  =  5/0°  +  ^/95°'     The  fourth  proportional  to  Ay  B,  C  is 


Art.  3.]       products  of  coplanar  vectors.  437 


(A-'£)C  =  '  ^,3+^.^^  1  5/5!  +  6/90° ! 

I     -j-  2 


1X4-2X3 


-6/0° +  5/90°} 


'  I''  +  2' 

=  13.4/0^+11.2/90°. 

Square  of  a  Binomial  of  Vectors. — If  A  -{-  B  denotes  a 
sum  of  non-successive  vectors,  it  is  entirely  equivalent  to  the 
resultant  vector  C.  But  the  square  of  any  vector  is  a  positive 
scalar,  hence  the  square  oi  A  -{-  B  must  be  a  positive  scalar. 
Since  A  and  B  are  in  reality  components  of  one  vector,  the 
square  must  be  formed  after  the  rules  for  the  products  of  rect- 
angular components  (p.  432).     Hence 

{A^By  =  {A^B){A+B), 

=  A'  +  AB  +  BA  +  B\ 

=  A'  +  B'  +  SAB  +  SBA  +  VAB  +  VBA, 

=  A'  +  B"  +  2SAB. 

This  may  also  be  written  in  the  form 

a^  -{-b^  -\-  2ab  cos  afi. 

But  when  A  -\-  B  denotes  a  sum  of  successive  vectors,  there 
is  no  third  vector  C  which  is  the  complete  equivalent ;  and  con- 
sequently we  need  not  expect  the  square  to  be  a  scalar  quan- 
tity. We  observe  that  there  is  a  real  order,  not  of  the  factors, 
but  of  the  terms  in  the  binomial;  this  causes  both  product 
terms  to  be  AB^  giving 

(A  +  BJ^  A'  +  2AB  +  B' 

=  A'+B'  +  2SAB  +  2Y  AB. 
The  scalar  part  gives  the  square  of  the  length  of  the  third 
side,  while  the  vector  part  gives  four  times  the  area  included 
between  the  path  and  the  third  side. 

Square  of  a  Trinomial  of  Coplanar  Vectors. — Let^  -\- B -\- 
C  denote  a  sum  of  successive  vectors.  The  product  terms  must 
be  formed  so  as  to  preserve  the  order  of  the  vectors  in  the  tri- 
nomial ;  that  is,  A  is  prior  to  B  and   C,  and  B  is  prior  to  C, 


438  VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChAP.  IX.. 

Hence 

(^  +^  +  C/  =A'-\-B'+0  +  2AB  +  2AC+  2BC, 

=  A'-\-B'  +  r-{-  2{SAB  -^SAC-\-  SBC),  (i) 
+  2{Y AB  -^VAC+  YBC).  {2) 

Hence  S{A -\- B -{- Cy  =  {i) 

—  ^»  _|_  ^»  _j-  ^a  _|_  2ab  COS  a/3  +  2ac  cos  ay  +  2bc  cos  /?;/ 
and  V(^  +  ^+Cy  =  (2) 

=  \2ab  sin  «/?  +  2ac  sin  or;/  +  2/^^  sin  fty\.aft 

The  scalar  part  gives  the  square  of  the  vector  from  the  be- 
Q     ginning  of  A  to  the  end  of  C  and  is  all  that  exists 
when  the  vectors  are  non-successive.     The  vector 
IQ  part  is  four  times  the  area  included  between  the 
successive   sides   and   the    resultant   side  of   the 
"^     "       polygon. 
Note  that  it  is  here  assumed  that  Y{A  -\- B)C  =  YAC-\- 
YBC,  which  is  the  theorem  of  moments.     Also  that  the  prod- 
uct terms  are  not  formed  in  cycHcal  order,  but  in  accordance 
with  the  order  of  the  vectors  in  the  trinomial. 

Example.— Let  A  =  3/a  B  =  5/30°,  C  =  7/45°  I  find  the 
area  of  the  polygon. 

^Y{AB+AC  +  BC), 
=  J|i5sin/o/30°  +  2i  sin/o/45°  +  35  sin/3oV45°}, 

=  3.75  +  742  +  4.53  =  15.7. 

Prob.  10.  At  a  distance  of  25  centimeters  /2o°  there  is  a  force 
of  1000  dynes  /8o°;  find  the  moment. 

Prob.  II.  A  conductor  in  an  armature  has  a  velocity  of  240 
inches  per  second  7300°  and  the  magnetic  flux  is  50,000  lines  per 
square  inch  /o;  find  the  vector  product. 

(Ans.   1.04  X  10^  Hnes  per  inch  per  second.) 

Prob.  12.  Find  the  sine  and  cosine  of  the  angle  between  the 
directions  0.8141  E.  +  0.5807  N.,  and  0.5060  E.  +  0.8625  N. 

Prob.  13.  When  a  force  of  200  pounds  7270°  is  displaced  by 
10  feet  730°,  what  is  the  work  done  (scalar  product)  ?  What  is  the 
meaning  of  the  negative  sign  in  the  scalar  product  ? 


Art.  4.]  COAXIAL    QUATERNIONS.  439 

Prob.  14.  A  mass  of  loo  pounds  is  moving  with  a  velocity  of  30 
feet  E.  per  second  -\-  50  feet  SE.  per  second;  find  its  kinetic  energy. 

Prob.  15.  A  force  of  10  pounds  745°  is  acting  at  the  end  of  8 
feet  /2oo°;  find  the  torque,  or  vector  product. 

Prob.  16.  The  radius  of  curvature  of  a  curve  is  2/0°  -f  5/90°; 
find  the  curvature.  (Ans.  .03/0°  -f  .17/90°.) 

Prob.  17.  Find  the  fourth  proportional  to  10/0°  -j-  2/90'' 
8/0°  -  3/9o_°,  and  6/o_°  +  s/90^.  ^         ' 

Prob.  18.  Find  the  area  of  the  polygon  whose  successive  sides 
are  10/30°,  9/100°,  8/180°,  7/225°. 

Art.  4.    Coaxial  Quaternions. 

By  a  "  quaternion  "  is  meant  the  operator  which  changes 
one  vector  into  another.  It  is  composed  of  a  magnitude  and 
a  turning  factor.  The  magnitude  may  or  may  not  be  a  mere 
ratio,  that  is,  a  quantity  destitute  of  physical  dimensions ;  for 
the  two  vectors  may  or  may  not  be  of  the  same  physical  kind. 
The  turning  is  in  a  plane,  that  is  to  say,  it  is  not  conical.  For 
the  present  all  the  vectors  considered  lie  in  a  common  plane  ; 
hence  all  the  quaternions  considered  have  a  common  axis."^ 

Let  A  and  R  be  two  coinitial  vectors;  the  direction  normal 
to   the   plane   may   be   denoted    by   /?.     The    operator   which 
changes  A  into  R  consists  of  a  scalar  multiplier         j^ 
and  a  turning  round  the  axis  /3.    Let  the  former  be 
denoted  by  r  and  the  latter  by  ,^^,  where  0  denotes 
the  angle  in  radians.      Thus  R  =  r/3^A  and  recip- 
rocally A  =  ~/3-'R.     Also  ~R  =  r/3'  and  ^A  =  -/3-^ 
^  r  A  ^  R  r 

The  turning  factor  /?^  may  be  expressed  as  the  sum  of  two 
component  operators,  one  of  which  has  a  zero  angle  and  the 
other  an  angle  of  a  quadrant.     Thus 

P^=cose  .  ^  -\-s\n6  .  y6f"/2. 

*  The  idea  of  the  "quaternion  "  is  due  to  Hamilton.  Its  importance  may 
be  judged  from  the  fact  that  it  has  made  solid  trigonometrical  analysis  possible. 
It  is  the  most  important  key  to  the  extension  of  analysis  to  space.  Etymologi- 
cally  "quaternion"  means  defined  by  four  elements;  which  is  true  in  space  •  in 
plane  analysis  it  is  defined  by  two. 


440  VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChAP.  IX. 

When  the  angle  is  naught,  the  turning-factor  may  be 
omitted ;  but  the  above  form  shows  that  the  equation  is 
homogeneous,  and  expresses  nothing  but  the  equivalence  of  a 
given  quaternion  to  two  component  quaternions.* 

Hence  r^^  =  r  cos  i9  -j-  ^  sin  d  .  ^/^ 

and  r^^A  =pA-\-  qft^/'A 

=^  pa  .  a-\-qa  .  ^'"f^a. 
The  relations  between  r  and  Q,  and/  and  ^,  are  given  by 


r=  ^p'-{-  q\    8  =  tan 

Example. — Let  E  denote  a  sine  alternating  electromotive 
force  in  magnitude  and  phase,  and  /  the  alternating  current  in 
magitude  and  phase,  then 

where  r  is  the  resistance,  /  the  self-induction,  n  the  alternations 
per  unit  of  time,  and  /3  denotes  the  axis  of  the  plane  of  repre- 
sentation.    It  follows  that  E  —  rl  -\-  2nnl ,  y^''/^/;  also  that 

that  is,  the  operator  which  changes  the  current  into  the  elec- 
tromotive force  is  a  quaternion.  The  resistance  is  the  scalar 
part  of  the  quaternion,  and  the  inductance  is  the  vector  part. 

Components  of  the    Reciprocal  of   a  Quaternion. — Given 
R={p-\-q'^'"')A, 
I 


then 


p-^q.^f^ 
p—  q  .  ^"'^ 

_p-q.^^l^ 
^ -ir-. 5—  K- 


R 


*  In  the  method  of  complex  numbers  y^Va  is  expressed  by  i,  which  stands 
for  y^—  I.  The  advantages  of  using  the  above  notation  are  that  it  is  capable 
of  being  applied  to  space,  and  that  it  also  serves  to  specify  the  general  turning 
factor  y5^  as  well  as  the  quadrantal  turning  factor  ySVa. 


Art.  4.]  coaxial  quaternions.  441 

Example. — Take  the  same  application  as  above.  It  is  im- 
portant to  obtain  /  in  terms  of  E.  By  the  above  we  deduce 
that  from  £  =  (r  +  271  nl ,  ^^/^)I 

I-  \  ^ ^^^^^  /3V2  I  E 

^  ~  \r'-\-(2nniy      r' ^ {27tnlY - '^       )  ^' 

Addition  of  Coaxial  Quaternions. — If  the  ratio  of  each  of 
several  vectors  to  a  constant  vector  A  is  given,  the  ratio  of 
their  resultant  to  the  same  constant  vector  is  obtained  by  tak- 
ing the  sum  of  the  ratios.     Thus,  if 

Rn  =   (A  +  ^n  .  /i"/^)^, 

then  2R-=.{ 2 />  +  {2g) .  ^-/']A, 

and  reciprocally 

_  Sp  -  (^y)  ■  0-' 

Example. — In  the  case  of  a  compound  circuit  composed 
of  a  number  of  simple  circuits  in  parallel 

_  r,-2  7tnl, . ^-/2  r, -2;r<.^-/^ 

^»  -   r,'  +  (27rnf/:     '  ''  ~    r,'  +  (2;r;/)V;  ^'    ^^^'' 

therefore,  ^/  =  2  \     ,   ,    . — -^  [  E 

[        \r  -\-{2  7tnyr}  r  -\-{2nnyP  '^      j 

and  reciprocally 

:s(  ,  ,  ,^     ,,„)  +  2  7r;/:^-(  ,  ,  /     ,,.J .  /5-/2 
_      \r'-\-{27tnfPi  \r'-\-{27tnyP/ 

Product  of  Coaxial  Quaternions. — If  the  quaternions  which 
change  A  to  R,  and  R  to  R',  are  given,  the  quaternion  which 
changes  A  to  /?'  is  obtained  by  taking'the  product  of  the  given 
quaternions. 

*This  theorem  was  discovered  by  Lord  Rayleigh;  Philosophical  Magazine, 
May,  1886.     See  also  Bedell  &  Crehore's  Alternating  Currents,  p.  238. 


442  VECTOR    ANALYSIS   AND    QUATERNIONS.  [Chap.  IX.. 

Given  R  =  rp^A  =z{p  +  q.  /3^/^)A 

and  R'  =  r'p^'R  =  {p'  +  q' .  fi-'^)R, 

then  R  =  r//3^+^'  A  ={^pp'  -  qq')  +  {pq'  +p'q) .  y^'^/^}^. 

Note  that  the  product  is  formed  by  taking  the  product  of 
the  magnitudes,  and  Hkewise  the  product  of  the  turning  fac- 
tors. Tlie  angles  are  summed  because  they  are  indices  of  the 
common  base  f^,^ 

Quotient  of  two  Coaxial  Quaternions. — If  the  given  qua- 
ternions are  those  which  change  A  to  R,  and  A  to  R\  then  that 
which  changes  R  to  R'  is  obtained  by  taking  the  quotient  of 
the  latter  by  the  former. 

Given    R  =  rjS^A  =  {p  +  q  .  ^''^')A 
and  R  =  /JS^'A  =  {p'  +  ^ .  ^-I^)A, 

then  R  ^~  fi^'-^R, 

r 

=  (/  +  ,'.  ^'A)^_^J-^ie, 

=  (/  +  /. ^v.)(/-^;^rV 

-~~ — 7+7 —^- 

Prob.  19.  The  impressed  alternating  electromotive  force  is  200 
volts,  the  resistance  of  the  circuit  is  10  ohms,  the  self-induction  is 
y^  henry,  and  there  are  60  alternations  per  second  ;  required  the 
current.  (Ans.   18.7  amperes /—_2o^^_42^.) 

Prob.  20.  If  in  the  above  circuit  the  current  is  10  amperes,  find 
the  impressed  voltage. 

Prob.  21.  If  the  electromotive  force  is  no  volts  /^  and  the  cur- 
rent is  10  amperes  /^  —  ^tt,  find  the  resistance  and  the  self-induc- 
tion, there  being  120  alternations  per  second. 

Prob.  22. ,  A  number  of  coils  having  resistances  /',,  ?\,  etc.,  and 
self-inductions  Z^,/,,  etc.,  are  placed  in  series  ;  find  the  impressed 
electromotive  force  in  terms  of  the  current,  and  reciprocally. 

*Many  writers,  such  as  Hayward  in  "Vector  Algebra  and  Trigonometry, " 
and  Stringham  in  "  Uniplanar  Algebra,"  treat  this  product  of  coaxial  quater- 
nions as  if  it  were  the  product  of  vectors.  This  is  the  fundamental  error  in  the 
Argand  method. 


Art.  5.]  addition  of  vectors  in  space.  445 

Art.  5.    Addition  of  Vectors  in  Space. 

A  vector  in  space  can  be  expressed  in  terms  of  three  inde- 
pendent components,  and  when  these  form  a  rectangular  set 
the  directions  of  resolution  are  expressed  by  i,j,  k.  Any  vari- 
able vector  R  may  be  expressed  as  R  ~-  rp  =  xi-\- yj -\-  zk,  and 
any  constant  vector  B  may  be  expressed  as 

B  =  b(i^b,i-\-bJ-\-bJi, 

In  space  the  symbol  p  for  the  direction  involves  two  ele- 
ments.    It  may  be  specified  as 

xi  -\-  yj  -{-  zk 

where  the  three  values  are  subject  to  the  condition  that  their 
sum  is  unity.  Or  it  may  be  specified  by  this  notation,  <p//0,. 
a  generalization  of  the  notation  for  a  plane.  The  additional 
angle  0/  is  introduced  to  specify  the  plane  in  which  the  angle 
from  the  initial  line  lies. 

If  we  are  given  R  in  the  form  rcp/fO,  then  we  deduce  the 
other  form  thus : 

R  =  r  cos  6  .i-\-  r  sm  6  cos  0 ./  -|-  r  sin  ^  sin  0  .  k. 
If  R  is  given  in  the  form  xi  -\- yj  -{-  2k,  we  deduce 


z    II  VV  4-  z^ 

i?  =  i/^'  +  /  -f  z^  tan-' - //  tan-    ^  J     . 


For  example,    B  =  10  3oV/45_ 

=  10  cos  45°.  t+  10  sin  45°  cos  30°  ./ +  10  sin  45°  sin  30°  .  k. 
Again,  from  C  -  3^  +  4J  +  S^  we  deduce 

5   //^     _,  I/4T 


C=   1/9  +  16  +  25  tan-'  -  //  tan 


=  7.07  5i°.4//64^9. 

To  find  the  resultant  of  any  number  of  component  vectors: 
applied  at  a  common  point,  let  R^,  R^,  .  .  .  R„  represent  the  /t 
vectors  or. 


444  VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChAP.  IX. 


R.  =;ir/  +  /,y+^,/^, 

R,  =^,i+yJ  +  ^.^^ 

Rn  =  ^J+ynJ-\-^nk\ 

en 

^R  =  (^x)i  +  (^>y+  {2z)k 

d 

r  =  v{^^y  +  {^yT  +  {^^y\ 

tan  0  : 

2^      ,  ,    .    V{^yy  +  (^^y 

-^>     and     tan^                 ^^ 

Successive  Addition. — When  the  successive  vectors  do  not 
lie  in  one  plane,  the  several  elements  of  the  area  enclosed  will 
lie  in  different  planes,  but  these  add  by  vector  addition  into  a 
resultant  directed  area. 

Prob.  23.  Express  ^  =  4/  —  5/'  +  6k  and  B  =  ^t  +  6/  —  jk  in 
the  form  r'^/^.      (Ans.  8.8  1307/63°  and  10.5  3'^7/6i°-5-) 

Prob.  24.  Express  C=  123  577/142°  and  Z>  =  456  657/200° 
in  the  form  xt  +yj  ^  zk.  

„             n  / /n        ,     ^                n  / /  n  . 
Prob.  2^.   Express  E  =  100  -  //  -    and   F  =  1000  -  //  3-   in 
^  4//   3  6// 4 

the  form  xt  -\-yj  +  2k.  

Prob.  26.  Find  the  resultant  of  10  20°//3o°,  20  307/4^,  and 

30  ^//S^'  __ 

Prob.  27.  Express  in  the  form  r(p//^  the  resultant  vector  of 
It  +  2/  —  3/^,  4/  —  5;*  +  6k,  and  —  7/  +  Sj  +  9/^. 

Art^  6.    Product  of  two  Vectors. 

Rules  of  Signs  for  Vectors  in  Space. — By  the  rules  t'  =  -\-, 
j^  z=  -^,  if  =  k,  andyV  =  —  k  we  obtained  (p. 432)  a  product  of 
two  vectors  containing  two  partial  products,  each  of  which  has 
the  highest  importance  in  mathematical  and  physical  analysis. 
Accordingly,  from  the  symmetry  of  space  we  assume  that  the 
following  rules  are  true  for  the  product  of  two  vectors  in  space  : 

if  —  k,  jk  =  i,  ki  =  y, 

ji  =  —  ^,  kj=^—  /,  ik  =  —J. 

The  square  combinations  give  results  which  are  indepen- 


Art.  6.]  PRODUCT  OF  two  vectors.  445 

dent  of  direction,  and  consequently  are  summed  by  simple 
addition.  The  area  vector  determined  by 
^*  and  yean  be  represented  in  direction  by  k^ 
because  k  is  in  tri-dimensional  space  the  axis 
which  is  complementary  to  /  and/.  We  also 
observe  that  the  three  rules  ij  =  k,  jk  —  i, 
ki  =y  are  derived  from  one  another  by  cyc- 
lical permutation ;  likewise  the  three  rules 
ji  ^=  —  k,  kj  =  —  iy  ik  =  —J,  The  figure  shows  that  these 
rules  are  made  to  represent  the  relation  of  the  advance  to  the 
rotation  in  the  right-handed  screw.  The  physical  meaning  of 
these  rules  is  made  clearer  by  an  appHcation  to  the  dynamo  and 
the  electric  motor.  In  the  dynamo  three  principal  vectors  have 
to  be  considered  :  the  velocity  of  the  conductor  at  any  instant, 
the  intensity  of  magnetic  flux,  and  the  vector  of  electromotive 
force.  Frequently  all  that  is  demanded  is,  given  two  of  these 
directions  to  determine  the  third.  Suppose  that  the  direction 
of  the  velocity  is  i,  and  that  of  the  flux/,  then  the  direction  of 
the   electromotive  force  is  k.     The   formula   ij  ■=  k  becomes 

velocity  flux  =  electromotive-force, 
from  which  we  deduce 

flux  electromotive-force  =  velocity, 
and  electromotive-force  velocity  =  flux. 

The  corresponding  formula  for  the  electric  motor  is 
current  flux  =  mechanical-force, 
from  which  we  derive  by  cyclical  permutation 

flux  force  =  current,     and     force  current  =  flux. 

The  formula  velocity  flux  =  electromotive-force  is  much 
handier  than  any  thumb-and-finger  rule  ;  for  it  compares  the 
three  directions  directly  with  the  right-handed  screw. 

Example. — Suppose  that  the  conductor  is  normal  to  the 
plane  of  the  paper,  that  its  velocity  is  towards  the  bottom,  and 
that  the  magnetic  flux  is  towards  the  left ;  corresponding  to 
the  rotation  from  the  velocity  to  the  flux  in  the  right-handed 
screw  we  have  advance  into  the  paper :  that  then  is  the  direc- 
tion of  the  electromotive  force. 

Again,  suppose  that  in  a  motor  the  direction  of  the  current 


446 


VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChAP.  IX. 


along  the  conductor  is  up  from  the  paper,  and  that  the  mag- 
netic flux  is  to  the  left ;  corresponding  to  current  flux  we  have 
advance  towards  the  bottom  of  the  page,  which  therefore  must 
be  the  direction  of  the  mechanical  force  which  is  appHed  to 
the  conductor. 

Complete  Product  of  two  Vectors. — Let^  =  ^i^  +  ^ay+^s'^ 
and  B  =  bJ.-{-b^j  -\-  b^k  be  any  two  vectors,  not  necessarily 
of  the  same  kind  physically.  Their  product,  according  to  the 
rules  (p.  444),  is 

AB  =  {a,i  +  aJ-\-  a,k){b,i  +  bj  +  b^k\ 

=  aJ)^ii-\-  ajb^jj  -{- aj^kk, 

+  ^Aj^  +  ^A^J  +  ^zb.ki  +  aj?^ik  +  a.bjLJ  +  aj)Ji 

=  a  J),  +  aj)^  +  aj)^, 


=  ^A  +  ^A  +  ^,<^8  + 


b. 


,  which  has  the  direction  normal  to  the  plane  of 


^3 

K 

i     j      k 

Thus  the  product  breaks  up  into  two  partial  products, 
-namely,  aJ}^-\-  aJ}^-\- aj)^^  which  is  independent  of  direction,  and 

^1       ^a       ^8 

K    K    K 

i     j     k 

A  and  B.     The  former  is  called  the  scalar  product,  and  the 
latter  the  vector  product. 

In  a  sum  of  vectors,  the  vectors  are  necessarily  homogene- 
ous, but  in  a  product  the  vectors  may  be  heterogeneous.  By 
making  a^  =  b^  =  o,  we  deduce  the  results  already  obtained 
for  a  plane. 

Scalar  Product  of  two  Vectors. — The  scalar  product  is  de- 
noted as  before  by  SAB.  Its  geometrical 
meaning  is  the  product  of  A  and  the  orthog- 
onal projection  of  B  upon  A.  Let  OP  rep- 
resent A,  and  OQ  represent  B,  and  let  OL, 
LM,  and  MN  be  the  orthogonal  projections 
upon  OP  of  the  coordinates  b^i,  bJ,  b^k  re, 
spectively.  Then  ON  is  the  orthogonal  pro- 
jection of  OQ,  and 


Art.  6.]  product  of  two  vectors.  447 

OP  X  ON  =  OP  X  (OL  +  LM  +  MN), 

\    a  a  a) 

=  aj)^  +  aj)^  4"  aj)^  —  SAB. 

Example.  —  Let  the  intensity  of  a  magnetic  flux  be 
B  =  bj,  ■\-  bj  -\-  bji,  and  let  the  area  be  5  =  sj,  -\-  sj  -\-  s^k  ; 
then  the  flux  through  the  area  is  SSB  =  b^s^  -\-  b^s^  -\-  b^s^. 

Corollary  i. — Hence  SB  A  =  SAB.     For 

b,a,  +  b,a,  +  b,a,  =  a,b,  +  aj)^  +  aj)^ . 

The  product  of  B  and  the  orthogonal  projection  on  it  of  A 
is  equal  to  the  product  of  A  and  the  orthogonal  projection  on 
it  of  B.  The  product  is  positive  when  the  vector  and  the  pro- 
jection have  the  same  direction,  and  negative  when  they  have 
opposite  directions. 

Corollary  2. — Hence  ^"  =  ^,'+^3" +^9'  =  ^"-  The  square  of 
A  must  be  positive  ;  for  the  two  factors  have  the  same  direction. 

Vector  Product  of  two  Vectors. — The  vector  product  as 
before  is  denoted  by  NAB.  It  means  the  product  of  A  and 
the  component  of  B  which  is  perpendicular  to  A^  and  is  rep- 
resented by  the  area  of  the  parallelogram  formed  by  A  and  B. 
The  orthogonal  projections  of  this  area  upon  the  planes  of  jk, 
ki,  and  ij  represent  the  respective  components  of  the  product. 
For,  let  OP  and  OQ  (see  second  figure  of  Art.  3)  be  the  or- 
thogonal projections  of  A  and  B  on  the  plane  of  i  and/;  then 
the  triangle  OPQ  is  the  projection  of  half  of  the  parallelogram 
formed  by  A  and  B.  But  it  is  there  shown  that  the  area  of 
the  triangle  OPQ  is  ^{afi^  —  ajb^.  Thus  {afi^  —ajb^k  denotes 
the  magnitude  and  direction  of  the  parallelogram  formed  by 
the  projections  of  A  and  B  on  the  plane  of  i  and/.  Similarly 
(^7/3  —  ajb^i  denotes  in  magnitude  and  direction  the  projec- 
tion on  the  plane  of  j  and  k,  and  (ajb^  —  aj)^)j  that  on  the 
plane  of  k  and  i. 

Corollary  i.— Hence  NBA  =  —  NAB. 

Example. — Given  two  lines  A  =^  ji  —  10;*+  ^k  and  B  = 
—  9^*  +  4/  —  dk  ;  to  find  the  rectangular  projections  of  the  par- 
allelogram which  they  define : 


448  VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChaP.  IX. 

NAB  =  (60  -  i2)i  +  (-  27  +  42);'+  (28  -  go)k 

=  48/  +  I  y  —  ^2k. 

Corollary  2. — If  A  is  expressed  as  aa  and  B  as  3/?,  then 
S^^  =  ab  cos  o'y^  and  YAB  =  ab  sin  a^  .  ory^,  where  ay^  de- 
notes the  direction  which  is  normal  to  both  a  and  /^,  and 
drawn  in  the  sense  given  by  the  right-handed  screw. 

Example.— Given  A  -  r^/e_  and  B  =  r'^//&_.     Then 

SAB  =  rr'  cos  ^/^07/^ 

=  rr'\Q,os  e  cos  6'  +  sin  ^  sin  6'  cos  (0'  —  0)}. 

Product  of  two  Sums  of  non-successive  Vectors. — Let  A  and 
B  be  two  component  vectors,  giving  the  resultant  A  -\-  B,  and 
let  C  denote  any  other  vector  having  the  same  point  of  appli- 
cation. 

Let  A  =  aj,  +  a  J  -\-  a^k^ 

■A+B 

B  =  b,i+bJ+b,k, 


Since  A  and  B  are  independent  of  order^ 
^  +  5  =:  («,  +  b:)i  +  K  +  b^j  +  {a,  +  b^k, 
consequently  by  the  principle  already  established 

^A  +  B)C  =  {a,  +  b,y,  +  {a,  +  b:)c,  +  {a,  +  b^ 

=  s^(::+s^c. 

Similarly  Y{A  +  B)C  =  \{a,  +  b^  -  K  +  ^yM+  etc. 
=  (^,c,  -  a,c,)t  +  ( V»  -  <^.^«)^'  +  •  •  • 
=  YAC+VBa 
Hence  (^  +  ^)(:  =  ^  C  +  ^C 

In  the  same  way  it  may  be  shown  that  if  the  second  factor 
consists  of  two  components,  C  and  D,  which  are  non-successive 
in  their  nature,  then 

{A+B){C+n)  =AC+AjD  +  BC  +  BI>, 


Art.  7.]  PRODUCT  OF  THREE  VECTORS.        *       449 

When  A  -{-  B  is  a.  sum  of  component  vectors 

{A  +  BY  =  A'  +  B'  +  AB  +  BA  ^  ^  _ 

=  A'  +  B'  +  2SAB. 

Prob.  28.  The  relative  velocity  of  a  conductor  is  S.W.,  and  the 
magnetic  flux  is  N.W.;  what  is  the  direction  of  the  electromotive 
force  in  the  conductor  ? 

Prob.  29.  The  direction  of  the  current  is  vertically  downward, 
that  of  the  magnetic  flux  is  West;  find  the  direction  of  the  mechani- 
cal force  on  the  conductor. 

Prob.  30.  A  body  to  which  a  force  of  2/  -^  y  -{-  4k  pounds  is 
applied  moves  with  a  velocity  of  5/+  67  +  'jk  feet  per  second;  find 
the  rate  at  which  work  is  done. 

Prob.  31.  A  conductor  8/+  0/  H~  10^  inches  long  is  subject  to 
an  electromotive  force  of  11/ +  127  +  13/^  volts  per  inch;  find  the 
difference  of  potential  at  the  ends.  (Ans.  326  volts.) 

Prob.  32.  Find  the  rectangular  projections  of  the  area  of  the 
parallelogram  defined  by  the  vectors  A  =  12/ —  23/—  34/^  and 
B=  ~  45/  -  56/'  +  67/^. 

•Prob.  33.  Show  that  the  moment  of  the  velocity  of  a  body  with 
respect  to  a  point  is  equal  to  the  sum  of  the  moments  of  its  com- 
ponent velocities  with  respect  to  the  same  point. 

Prob.  34.  The  arm  is  9/ -f-  ly  -|-  13^  feet,  and  the  force  applied 
at  either  end  is  17/ -f-  197  +  23k  pounds  weight;   find  the  torque. 

Prob.  35.  A  body  of  1000  pounds  mass  has  linear  velocities  of  5a 
feet  per  second  3oV/45^,  and  60  feet  per  second  6o°//2  2°.5;  find 
its  kinetic  energy. 

Prob.  36.  Show  that  if  a  system  of  area-vectors  can  be  repre- 
sented by  the  faces  of  a  polyhedron,  their  resultant  vanishes. 

Prob.  37.  Show  that  work  done  by  the  resultant  velocity  is  equal 
to  the  sum  of  the  works  done  by  its  components. 

Art.  7.    Product  of  Three  Vectors. 

Complete    Product.— Let    us    take   A  =  a  J,  +  a  J  +  afi, 
B  =  d,t  +  dj  +  d,k,  and  C  =  cj  +  cj+  c,k.     By  the  product 
of  A,  B,  and  C  is  meant  the  product  of  the  product  of  A  and 
B  with  Ct  according  to  the  rules    p.  444).     Hence 
ABC  =  {a,b,  +  a  A  +  ciA){c.i  +  cJ-\-c,k) 


430 


VECTOR    ANALYSIS   AND    QUATERNIONS. 


[Chap.  IX. 


+ 


a,  a. 

a,a^ 

a,  a^ 

Kb^ 

KK 

K  K 

c,             c^             c^ 

i 

J 

k      1 

(2)  + 


^1 

a. 

^3 

K 

K 

^3 

^1 

c^ 

<^3 

(3) 


Example.— Let  A  =  li -\-  2j-\-  ik,  B  =  41  +  5/  +  6k,  and 
C  =  yt  +  Sj-{-  gk.    Then 

(I)  =  (4  +  10  +  iS)(7t  +  Sj  +  gk)  =  22{7i+Sj  +  gk). 

ySt  +  6j  —  66k, 


(2)  = 

-3    6 

-3 

7    8       9 

i    j      k 

(3)  = 

I     2    3 
4    5    6 
;    8    9 

=  0. 

(3) 


If  we  write  yi  =  aa,  B  =  b/3,  C  =  cy,  then 
ABC  =  abc  cos  afi  .  y 

-\-  abc  sin  a^  sin  ay^y .  ay^;/ 
+  abc  sin  o'y^  cos  a^y^ 
where  cos  o^/  denotes  the  cosine  of  the  angle  between  the 
directions  a^  and  y,  and  a^y  denotes  the  direction  which  is 
normal  to  both  a^  and  y. 
We  may  also  write 

ABC='^AB.C-{-Ni^AB)C^'S^^AB)C, 
(I)  (2)  (3) 

First  Partial  Product. — It  is  merely  the  third  vector  multi- 
plied by  the  scalar  product  of  the  other  two,  or  weighted  by 
that  product  as  an  ordinary  algebraic  quantity.  If  the  direc- 
tions are  kept  constant,  each  of  the  three  partial  products  is 
proportional  to  each  of  the  three  magnitudes. 

Second  Partial  Product. —  The  second  partial  product  may 
be  expressed  as  the  difference  of  two  products  similar  to  the 
first.     For 

N{SAB)C^\-  (b^c^  +  ^3^3K  +  (^,^,  +  c,a,)bM 
+  1  -  (b.c,  +  b,c)a,  +  {c,a,  +  c,a,)b,\j 
+  !  -  (b,c^  +  b,cM,  +  {c,a,  +  c^a^b,}k. 


Art.  7.] 


PRODUCT  OF  THREE  VECTORS. 


451 


By  adding  to  the  first  of  these  components  the  null  term 
{b^c^a^  —  c^ajf^i  we  get  —  SBC.  aJ-{-  SCA  .  dj,  and  by  treating 
the  other  two  components  similarly  and  adding  the  results  we 

obtain 

V(VAB)C  =  -  SBC .  A  +  SCA .  B. 

The  principle  here  proved  is  of  great  use  in  solving  equa- 
tions (see  p.  455). 

Example. — Take  the  same  three  vectors  as  in  the  preced- 
ing example.     Then 

V{VAB)C  = -(2S  + 40 +S4){ii  + 2/ i-sk) 
+  (7  +  i6  +  27)(4^-+5;-  +  6^) 
=:  ySt -}- 6/ —  66k. 
The  determinant  expression  for  this  partial  product  may 
also  be  written  in  the  form 


a,  a. 


^1  <^, 
i    j 


+ 


K  K 


J    k 


+ 


b.  b. 


It  follows  that  the  frequently  occurring  determinant  expression 


a,  a, 

b.  b. 


+ 


KK 


dj. 


+ 


b^b. 


d,d. 


means  S(VAB){VCI?). 

Third  Partial  Product. — From  the  determinant  expression 
for  the  third  product,  we  know  that 

S(V^^)C:=  S(V^O^  =  S(VCA)B 
=  _  S(V^^)C:  =  -  S{VCB)A  =  -  S(VAC)B. 
Hence  any  of  the  three  former  may  be  expressed  by  SABC, 
and  any  of  the  three  latter  by  —  SABC. 

The  third  product  S{yAB)C  is  represented  by  the  vol- 
ume of  the  parallelepiped  formed  by  the  vectors  A,  B,  C 
taken  in  that  order.  The  line  YAB  v^b 
represents  in  magnitude  and  direction 
the  area  formed  by  A  and  By  and  the 
product  of  YAB  with  the  projection 
of  C  upon  it  is  the  measure  of  the 
volume  in  magnitude  and  sign.  Hence  the  volume  formed 
by  the  three  vectors  has  no  direction  in  space,  but  it  is  posi- 
tive or  negative  according  to  the  cyclical  order  of  the  vectors. 


452  VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChAP.  IX. 

In  the  expression  abc  sin  a^  cos  a^y  it  is  evident  that  sin  a^' 
corresponds  to  sin  B^  and  cos  afty  to  cos  0,  in  the  usual  for- 
mula for  the  volume  of  a  parallelepiped. 

Example. — Let  the  velocity  of  a  straight  wire  parallel  to- 
itself  be  F  =  1000/30°  centimeters  per  second,  let  the  intensity 
of  the  magnetic  flux  be  ^  =  6000  /qo"  lines  per  square  cen- 
timeter, and  let  the  straight  wire  L  —  \^  centimeters  60°/  /45°. 
Then  YVB  =  6000000  sin  60°  90°/  /90°  lines  per  centimeter  per 
second.  Hence  S{Y  VB)L  =  15  X  6000000  sin  60°  cos  0  lines- 
per  second  where  cos0  =  sin  45°  sin  60°. 

Sum  of  the  Partial  Vector  Products. — By  adding  the  first 
and  second  partial  products  we  obtain  the  total  vector  product 
of  ABC,  which  is  denoted  by  Y{ABC).  By  decomposing  the- 
second  product  we  obtain 

Y{ABC)  =  SAB.  C  -  SBC .  A  +  SCA  .  B. 
By  removing  the  common  multiplier  abcy  we  get 

Y(a/3y)  =  cos  aj3  .  y  —  cos  /3y .  a  -\-  cos  ya  ,  fi. 
Similarly  Y{/3ya)  =  cos  /3y  .  a  —  cos  ya .  /3  -\-  cos  a/3  .  y 
and  Y{ya^)  =  cos  ya  .  J3  —  cos  a/3  .  y  -^  cos  /3y  .  a. 

These   three  vectors   have  the   same   magnitude,  for  the 
square  of  each  is 
cos'  a^  4-  cos'  ^y  -f  cos'  ya  —  2  cos  a/3  cos  /3y  cos  ya, 
y/  that  is,  I  -^{S(a/3y)\\ 

They  have  the  directions  respectively  of  a\ 
la   ^'y  y'^  which  are    the  corners  of    the    triangle 
whose   sides    are    bisected    by   the  corners   a, 
^,  y  of  the  given  triangle. 

Prob.  38.  Find  the  second   partial  product  of 

9  207/30°,  10  307/40°,  II  457/45!-  Also  the  third  partial 
product. 

Prob.  39.  Find  the  cosine  of  the  angle  between  the  plane  of 
IJ^m^j-^n^k3,nd.  lJ-\-m^j-\-n^k  and  the  plane  oi  IJ -\- mj -\- n^k. 
and  I  J  -\-  mJ-\-  nji. 

Prob.  40.  Find  the  volume  of  the  parallelepiped  determined  by 
the  vectors  ioo/+5q;"  +  25/^,  50/+ ioy+8o/^,  and  -  75/+4oy-8o>^. 


Art.  8.]  COMPOSITION  of  quantities.  453 

Prob.  41.  Find  the  volume  of  the  tetrahedron  determined  by  the 
-extremities  of  the  following  vectors  :  3/  —  2/  -j-  ik,  —  4/  +  5/'  —  jky 
.3/  —  77  —  2^,  8/  +  4;'  —  3/^. 

Prob.  42.  Find  the  voltage  at  the  terminals  of  a  conductor  when 
its  velocity  is  1500  centimeters  per  second,  the  intensity  of  the  mag- 
netic flux  is  7000  lines  per  square  centimeter,  and  the  length  of  the 
conductor  is  20  centimeters,  the  angle  between  the  first  and  second 
being  30°,  and  that  between  the  plane  of  the  first  two  and  the  direc- 
tion of  the  third  60°.  (Ans.  .91  volts.) 

Prob.  43.  Let  a  =  ^/io°,  /3  =  ^725°,  y  =  4^/35°-  Find 
ycx/3y,  and  deduce  Y/3ya  and  Yya/3. 


Art.  8.    Composition  of  Quantities. 

A  number  of  homogeneous  quantities  are  simultaneously 
located  at  different  points ;  it  is  required  to  find  how  to  add  or 
compound  them. 

Addition  of  a  Located  Scalar  Quantity. — Let  m^  denote  a 
mass  m  situated  at  the  extremity  of  the  radius- 
vector  A.     A  mass  m  —  m  may  be  introduced 
at  the  extremity  of  any  radius-vector  Ry  so 

that 

m^  =  (m  —  ni)R  +  m^ 

=  mR  +  mA  —  Mr 

=  niR  +  m{A  —  R), 
Here  ^  —  7?  is  a  simultaneous  sum,  and  denotes  the  radius- 
vector  from  the  extremity  of  R  to  the  extremity  of  A.  The 
product  m{A  —  R)  is  what  Clerk  Maxwell  called  a  mass-vector, 
and  means  the  directed  moment  of  m  with  respect  to  the  ex- 
tremity of  R,  The  equation  states  that  the  mass  m  at  the 
extremity  of  the  vector  A  is  equivalent  to  the  equal  mass  at 
the  extremity  of  R,  together  with  the  said  mass-vector  applied 
at  the  extremity  of  R.  The  equation  expresses  a  physical  or 
mechanical  principle. 

Hence  for  any  number  of  masses,  w,  at  the  extremity  of  -^,, 
m^  at  the  extremity  of  A^,  etc., 

:SmA  =  ^niR  +  ^\m{A  —  R)\, 


454  VECTOR    ANALYSIS    AND    QUATERNIONS.  [Chap    IX. 

where  the  latter  term  denotes  the   sum  of  the  mass-vectors 
treated  as  simultaneous  vectors  applied  at  a  common  point. 

Since  :2\m{A  —  R)\  =  ^mA  -  :2mR 

=  ^niA  —  R'2m, 

the  resultant  moment  will  vanish  if 

R  =  —^, — ,       or       K27n  =  2imA 

Corollary.— Let       R  z=  xt -{-  yj  +  zk, 
and  A  —a,i  +  bj  -\-c,k', 

then  the  above  condition  may  be  written  as 
:E\m{ai  -\-bj-\-ck)} 

'    -'•'     '  ^711 

__  2 (ma)  .  t      (2f/ib)  .J      2(mc)  .  k 

^771  2771  277t       * 

2{7nd)  2(7nb)  27nc 

therefore  x  =  -^-^ — ,     y  —  -^^ —  ,     z  =  — — 

^771  -^  ^771  2m 

Example. — Given    5   pounds   at    10  feet  45y/30°  and    S 


pounds  at  7  feet  6o°//45°  I  ^^^^  the  moment  when  both  masses 

are  transferred  to  12  feet  75°//6o°. 

Tn^A^  =  5o(cos  30°/  +  sin  30°  cos  45°/+  sin  30°  sin  45 °>^), 
m^A^  =  56(cos  45°^  +  sin  45°  cos  6o°y  +  sin  45°  sin  60° k), 

{m^  +  Tn^R  —  i26(co.'>  60°/  +  sin  60°  cos  75*^'  +  sin  60°  sin  75°/^), 

moment  =  7n^A^  +  7n^A^  —  {ni^  -\-  77i^R. 

Composition   of  a  Located  Vector  Quantity. — Let  Fa  de- 
note a  force  applied  at  the  extremity  of  the  radius- vector  A. 
As  a  force  F  —  F  may  introduced  at  the  ex- 
tremity  of  any  radius-vector  R,  we  have 


Fa  =  {F-F),  +  Fa 
=  F,  +  V{A-R)F 

This  equation  asserts  that  a  force  F  applied 
at  the  extremity  of  A  is  equivalent  to  an  equal  force  applied' 
at  the  extremity  of  R  together  with  a  couple  whose  magnitude 


Ar.T.  8.]  COMPOSITION    OF    QUANTITIES.  455 

and  direction  are  given  by  the  vector  product  of  the  radius- 
vector  from  the  extremity  of  R  to  the  extremity  of  A  and  the 
force. 

Hence  for  a  system  of  forces  appHed  at  different  points, 
such  diS  F^^X  A^y  F^dX  A^,  etc.,  we  obtain 

2{F^)  =  2{F^)  +  :^Y(A  -  R)F 
=  {:SF),  +  :SY{A-R)F 

Since  2Y{A  -  R)F  =  2VAF  -  2VRF 

=  ^VAF-VR^F 
the  condition  for  no  resultant  couple  is 

YR2F=:^VAF, 
which  requires  2F  to  be  normal  to  '2V AF, 

Example. — Given  a  force  u*-|-2/  +  3^  pounds  weight  at 
4^  +  Sj-\-^^  ^^^t,  and  a  force  of  7/ +  9;*+  ii>^  pounds  weight 
at  loi -{- I2j -{- i/^k  feet;  find  the  torque  which  must  be  sup- 
plied when  both  are  transferred  to  2^' -["  5/ +  S'^*  so  that  the 
eflect  may  be  the  same  as  before. 

WA^F,=^6i-  \2j-^6k, 
2YAF=gi-i^j+gk, 
:2F=%i-\.iij-{-i^^ 
YR2F=37t-4j-iSk, 
Torque  =  —  28/—  147  +  ^yk. 
By  taking  the  vector  product  of  the  above  equal  vectors 
with  the  reciprocal  of  ^F  we  obtain 

By  the  principle  previously  established  the   left   member 

resolves  into  —  R -{- SR-^^.  2F;   and    the  right   member   is 

equivalent  to  the  complete  product  on  account  of  the  two 
factors  being  normal  to  one  another ;  hence 


456  VECTOR    ANALYSIS   AND    QUATERNIONS.  [ChAP.  IX. 

that  is,  R  =  -^^QJAF)  +  SR^ .  2F. 

(I)  (2) 

The  extremity  of  R  lies  on  a  straight  Hne  whose  perpen- 
dicular is  the  vector  (i)  and  whose  direction  is  that 
of  the  resultant  force.  The  term  (2)  means  the 
projection  of  R  upon  that  hne. 

The  condition  for  the  central  axis  is  that  the 
resultant    force    and    the    resultant    couple   should 
have  the  same  direction  ;  hence  it  is  given  by 
V j 2WAF -  YR:EF\ 2F=0] 
that  is,  V{YR^F)2F  =  Y(:SAF)2F 

By  expanding  the  left  member  according  to  the  same  prin- 
ciple as  above,  we  obtain 

—  {:^FyR  +  SR2F.  2F=  Y(2AF)2F; 

therefore  H  =  -^^V^F{V2AF)  +  j^^ .  2F 

This  is  the  same  straight  line  as  before,  only  no  relation  is 
now  imposed  on  the  directions  of  2F  and  2VAF;  hence  there 
always  is  a  central  axis. 

Example. — Find  the  central  axis  for  the  system  of  forces 
in  the  previous  example.  Since  2  F=  St-{- iij-\-' 14k,  the 
direction  of  the  line  is 

Sz-\-  iJj-{-  14k 

V64  +  121  +  196* 

Since  ^  =  {  and  2YAF  =  9/  -  1 8/  +  gk,  the 

2,P  351 

perpendicular  to  the  line  is 

.,  8^  +  ii/+i4/^   .        ^.  .      ,         I    ,        .  .         .  ,, 

V-  ^  ^^^  ^-9^-i8;  +  9^=:3— J35i^  +  54y-243>^j. 

Prob.  44.   Find  the  moment  at  qoV/27o°  of  10  pounds  at  4  feet 

io°//2o°  and  20  pounds  at  5  feet  30°//! 20°. 


Art.  9.]  SPHERICAL    TRIGONOMETRY.  457 

Prob.  45.  Find  the  torque  for  4/ +37'+  2k  pounds  weight  at 
2^'—  3/' +  1^  feet,  and  2/—  ly  —  i>^  pounds  weight  at  —  3/+  4/'-f  ^k 
feet  when  transferred  to  —  3/  +  2/  —  /^k  feet. 

Prob.  46.  Find  the  central  axis  in  the  above  case. 

Prob.  47.  Prove  that  the  mass-vector  drawn  from  any  origin  to  a 
mass  equal  to  that  of  the  whole  system  placed  at  the  center  of  mass 
of  the  system  is  equal  to  the  sum  of  the  mass-vectors  drawn  from 
the  same  origin  to  all  the  particles  of  the  system. 

Art.  9.    Spherical  Trigonometry. 

Let  2,/,  k  denote  three  mutually  perpendicular  axes.  In 
order  to  distinguish  clearly  between  an  axis  and  a  quadrantal 
version  round  it,  let  ^     ,y     ,  ^    denote  ^  fe 

quadrantal  versions  in  the  positive  sense 
about   the  axes  i^j\  k  respectively.     The 
directions  of  positive  version  are  indicated   ~^j 
by  the  arrows. 

By  t'^H'"^'^  is  meant  the  product  of  two 
quadrantal  versions  round   i\  it  is  equiv-  ^^ 

alent  to  a  semicircular  version  round  i]  hence  r^H'"^''  =  t"  =—. 
Similarly y^y"'^'  means  the  product  of  two  quadrantal  versions 
roundy,  and/V^»  =>"  =  ~.     Similarly  ^Z^^"/'  =  r  =  -. 

By  t'^'^f^'^  is  meant  a  quadrant  round  i  followed  by  a  quad- 
rant round/;  it  is  equivalent  to  the  quadrant  fromy  to  i,  that 
is,  feo  —  ]i''\  Buty^T'^''  is  equivalent  to  the  quadrant  from  —  i 
to  — y,  that  is,  to  H'''',  Similarly  for  the  other  two  pairs  of 
products.     Hence  we  obtain  the  following 

Rules  for  Versors. 

The  meaning  of  these  rules  will  be  seen  from  the  follow 
ing  application.      Lei    li  -\-  mj  -f-  nk  denote    any  axis,  then 


458  VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChAP.  IX, 

(// _|_  ^/^y  _|_  nky/"^  denotes -a  quadrant  of  angle  round  that  axis. 
This  quadrantal  version  can  be  decomposed  into  the  three 
rectangular  components  /r^',  mf'"",  nH'''' ;  and  these  components 
are  not  successive  versions,  but  the  parts  of  one  version.  Sim- 
ilarly any  other  quadrantal  version  {I'i  -\-  m'j  -\-  ri'kf'''  can  be 
resolved  into  IT^^  m'f''',  n'k''^\  By  applying  the  above  rules, 
we  obtain 
(//  _|-  ^nfj^  nk)"'\l'i  +  m'j  +  n'kf'^ 

=  (^li^/^  4_  ^f/^  _|-  nk^'^'Wi''^'  +  m'f'^  +  n'k"'^) 

=z  —  [H ' -\- mm' -{- nn') 

-  {mn'  -  m'n)^^''  -  {nl'  -  n'J)f'^  -  (Im'  -  l'm)k'''' 
z=—{ll'+mm'-\-nn') 

-  {{mn'  -  m'n)i-{-  {nl'  -  n'l)j'\-  {Im'  -  l'm)kY'\ 

Product  of  Two  Spherical  Versors. — Let  yS  denote  the  axis 
and  b  the  ratio  of  the  spherical  versor  PA,  then  the  versor 
itself  is  expressed  by  yS*.  Similarly  let  y 
denote  the  axis  and  c  the  ratio  of  the 
spherical  versor  AQ,  then  the  versor  itself 
is  expressed  by  y". 

Now     /?*  =  cos  ^  +  sin  b .  fT'^ 
and  y'  =  cos  ^  +  sin  ^  .  y"^^ ; 

therefore 
/3y  =  (cos  b-\-smb.  /r/»)(cos  ^  +  sin  <: .  y'^') 

=  cos  b  cos  c  -\-  cos  ^  sin  ^ .  y'^^'  +  cos  ^  sin  3 .  /3'^* 

-\-  sin  <^  sin  ^ .  /S^^'y. 
But  from  the  preceding  paragraph 

p-/.y^/.    ^    _    cos  ^^  _   sJj^  ^^  ^   ^tA  . 

therefore        /3^y^  =  cos  b  cos  r  —  sin  ^  sin  c  cos  /3y  (i) 

-j-  \  cos  b  sine  .y  -\-  cos  ^  sin  3 .  y^  —  sin  b  sin  c  sin  /3y  .  ^y  \  ^^*.  (2) 

The  first  term  gives  the  cosine  of  the  product  versor  ;  it  is 
equivalent  to  the  fundamental  theorem  of  spherical  trigonom- 
etry, namely, 

cos  a  =  cos  b  cos  ^  +  sin  <^  sin  ^  cos  A , 


Art,  9.]  SPHERICAL    TRIGONOMETRY.  459 

where  A   denotes  the  external  angle  instead  of  the  angle  in- 
cluded by  the  sides.  -  - 

The  second  term  is  the  directed  sine  of  the  angle ;  for  the 
square  of  (2)  is  equal  to  I  minus  the  square  of  (i),  and  its  di- 
rection is  normal  to  the  plane  of  the  product  angle.* 


Example.— Let  /?  =  307/45°  and  y  =  607/30°.     Then 
cos  /3y  =  cos  45°  cos  30°  +  sin  45°  sin  30°  cos  30°, 
and  sin  /3y  .  Py  ==  VPy ; 

but  /3  =  cos  45°  t  +  sin  45°  cos  30°/+  sin  45°  sin  30°  ky 
and  y  =  cos  30°  t  +  sin  30°  cos  6o°y-|-  sin  30°  sin  60°  k ; 
therefore 

VPy  =  jsin  45°  cos  30°  sin  30°  sin  60° 

—  sin  45°  sin  30°  sin  30°  cos  60°  j 2 
+  jsin  45°  sin  30°  cos  30°  —  cos 45°  sin  30°  sin  60° f/ 
+  J  cos  45°  sin  30°  cos  60°  —  sin  45°  cos  30°  cos  30°  |/^. 

Quotient  of  Two  Spherical  Versors. — The  reciprocal  of  a 
given  versor  is   derived    by  changing  the  sign  of  the  index ; 
y'"  is   the    reciprocal  of  y^     As  /5*  =  cos  ^  -|-  sin  d .  /S^'^'y  and 
y-^  =  cos  c  —  sin  c .  y"^  ', 
/S^y'"  =  cos  d  cos  ^  +  ^in  ^  sin  c  cos  fiy 

+  ]cos  ^  sin  d,  /3—  cos  ^  sin  c.  y -{- sin  d  sin  c  sin  /3y »  /3y  j-"^'- 

S! 

Product  of  Three  Spherical  Versors. — Let 
«*  denote  the  versor  PQ,  /J*  the  versor  QR, 
and  y^  the  versor  RS ;  then  a'^p^y"  denotes 
PS.     Now  a'^P^'y'  ^^ 

=  (cos  ^  + sin  ^.o'''/')(cos<^4-sin^./?''/»)(cos^  +  sin  c .  y''''') 
=  cos  a  cos  l^  cos  c  (i) 

-\-  cos  a  cos  ^  sin  c.  y""^^  +  cos  a  cos  ^  sin  ^ .  /5 

-|-  cos  d  cos  ^  sin  ^ .  a  (2) 

-\-  cos  a  sin  ^  sin  ^ .  p^^'^y"''^  -f-  cos  ^  sin  <a;  sin  ^ .  or   '';/ 

+  cos  csm  as\nb .  a^'^ff''^     (3) 
*  Principles  of  Elliptic  and  Hyperbolic  Analysis,  p.  2. 


460  VECTOR    ANALYSIS    AND    QUATERNIONS.  [Chap.  IX. 

+  sin  a  sin  bsixic,  a''^'/r^y''^\  (4) 

The  versors  in  (3)  are  expanded  by  the  rule  already  ob- 
tained, namely, 

^-/y  V,  ^  _  cos  fty  -  sin  ^y .  Jy^\ 

The  versor  of  the  fourth  term  is 

^-A^Vy-A  ^  _  (cos  ary^  +  sin  aft.  '^ff'^'^y'^ 

—  —  cos  aft  .  y''^'-{-sin  aft  cos  ary5/+sin  aft  sin  afty ,  afty""^'. 

Now  sin  aft  sin  afty .  afty  =  cos  ay  .  ft  —  cos  fty  .  a  (p.  45 1), 
hence  the  last  term  of  the  product,  when  expanded,  is 

sin  a  sin  ^  sin  <;{  -  cos  aft .  /^^  +  cos  ay .  ^-/» 

—  cos  fty .  a'^  +  Qoia~fty  \ . 
Hence 
cos  a'^ft^y^  =  cos  t?  cos  b  cos  ^  —  cos  a  sin  ^  sin  c  cos  /?;/ 

—  cos  b  sin  ^  sin  t  cos  ay  —  cos  ^  sin  a  sin  ^  cos  aft 
+  sin  ^  sin  b  sin  ^  sin  <^y5  cos  afty^ 

and,  letting  Sin  denote  the  directed  sine. 

Sin  a'^ft^y^  =  cos  a  cos  b  sine  .y  -{-  cos  ^  cos  r  sin  ^ .  /3 

-\-  cos  ^  cos  ^  sin  <^  .  a  —  cos  a  sin  ^  sin  c  sin  /?;/  .  y^;/ 

—  cos  b  sin  ^  sin  c  sin  or;/ .  ay 

—  cos  ^  sin  a  sin  ^  sin  «/? .  aft 

—  sin  <?  sin  ^  sin  ^5 cos  <ary^.  y  — cos  ay  .  ft-\-cos  fty ,  a}.* 

Extension  of  the  Exponential  Theorem  to  Spherical  Trigo- 
nometry.— It  has  been  shown  (p.  458)  that 

cos  ft^y"  =  cos  b  cos  ^  —  sin  ^  sin  c  cos  fty 
and 

<sin  ftyy^'  =  cos  ^  sin  ^ .  ft''^'  +  cos  ^  sin  ^  .  y""^' 

—  sin  b  sin  c  sin  fty .  fty^^'* 

Now  cos  ^=  I V -\- —T  —  zrr -{-  etc. 

2!    '4!       6! 

*  In  the  above  case  the  three  axes  of  the  successive  angles  are  not  perfectly 
independent,  for  the  third  angle  must  begin  where  the  second  leaves  off.  But 
the  theorem  remains  true  when  the  axes  are  independent  ;  the  factors  are  then 
quaternions  in  the  most  general  sense. 


Art.  9.]  SPHERICAL    TRIGONOMETRY.  461' 

and  s\x\b  =  b  —  —7--\-—7  —  etc. 

Substitute  these  series  for  cos  b,  sin  b,  cos  r,  and  sin  ^  in 
the  above  equations,  multiply  out,  and  group  the  homogeneous 
terms  together.     It  will  be  found  that 

cos  ^^y'  =  I  —  —{b''  +  2bc  cos  ^y  -{-  c"] 

+  ^1  /5*  +  ^b'c  cos  py  +  6^^^^  +  4^<;^  cos  py -\- c'\ 

-  ^{^'  +  6^V  cos  py  +  I5/5V'  +  20^V  cos  §y 

+  iS^V  +6/^^^  cos  ^yJ^c'\-^.,  ., 
where  the  coefficients  are  those  of  the  binomial  theorem,  the 
only  difference  being  that  cos  ^y  occurs  in  all  the  odd  terms 
as  a  factor.  Similarly,  by  expanding  the  terms  of  the  sine,  we 
obtain 
(Sin ^'fY'''  =  b,lf^'-[-c.  //  -  be  sin ^y .  J/^' 

+  -,-  \  be'  +  b'c  \sm0y.  ^''^ 

+  J^I^'.^j'A  +  S^Jv.  /"^  lob'c\f" 

+  lob'c'  ■  y""'  +  Sbc\  ft'"  +  c' .  r'"  \ 

By  adding  these  two  expansions  together  we  get  the  ex- 
pansion for  ft''y',  namely, 
fty=^  +  b.ft"'  +  c.y''" 

-  -V  ]  *'  +  2*<cos  fir  +  sin  fty .  ^"'0  +  c-  \ 


-^\b\p''f'^lb\,/''^lbc\^'"'^c\/"\ 

J^^\b'-\- 4(^vCcos  ^y  +  sin  py  .  ^^"^0  +  6/^V' 

+  4^-^'(cos  fiy  +  sin  Py  .  ^  '^'0  +  ^1  +  •  •  ► 


462  VECTOR    ANALYSIS    AND    QUATERNIONS.  [Chap.  IX. 

By  restoring  the   minus,  we  find   that    the  terms   on    the 
second  Hne  can  be  thrown  into  the  form 


and  this  is  equal  to 


^\b.(i"^-^c./'r, 


where  we  have  the  square  of  a  sum  of  successive  terms.     In  a 
similar  manner  the  terms  on  the  third  Hne  can  be  restored  to 

that  is,  -^{b.^  +  cy'^^'V, 

Hence 

Extension  of  the  Binomial  Theorem. — We  have  proved 
above  that  ^^^'^/^^v"/' = /^''^'  +  ^^'^/2  provided  that  the  powers 
of  the  binomial  are  expanded  as  due  to  a  successive  sum,  that 
is,  the  order  of  the  terms  in  the  binomial  must  be  preserved. 
Hence  the  expansion  for  a  power  of  a  successive  binomial  is 
given  by 
j^  .  iff^  -t_  c  .  //'}"  =  (^«  .  /5«"/'  +  nd'^-'c  .  y^^'-X'^/^y/^ 

+  ^y— ^^«-V.y5(«-'W2y-+etc. 

*  At  page  386  of  his  Elements  of  Quaternions,  Hamilton  says:  "In  the 
present  theory  of  diplanar  quaternions  we  cannot  expect  to  find  that  the  sum  of 
the  logarithms  of  any  two  proposed  factors  shall  be  generally  equal  to  the 
logarithm  of  the  product;  but  for  the  simpler  and  earlier  case  of  coplanar 
quaternions,  that  algebraic  property  may  be  considered  to  exist,  with  due 
modification  for  multiplicity  of  value."  He  was  led  to  this  view  by  not  dis- 
tinguishing between  vectors  and  quadrantal  quaternions  and  between  simul- 
taneous and  successive  addition.  The  above  demonstration  was  first  given  in 
my  paper  on  "The  Fundamantal  Theorems  of  Analysis  generalized  for  Space.'* 
It  forms  the  key  to  the  higher  development  of  space  analysis. 


Art.  10.]  COMPOSITION  OF  rotations.  463 

Example.— Let  b  =  j\  and  c=i,  /3  =  W/MS°,  Y  =  607/30°. 
(^ .  yS-/2  _^  c .  y'/'y  =  -\d'+c'-i-  2bc  cos  §y  +  2Msin  y6f/)*'^'J 

=  -  (xk  +  ^V  +  3^  cos  ^y)  -  -i^{sm  pyf\ 
Substitute  the  calculated  values  of  cos  ^y  and  sin  fiy. 

Prob.  48.  Find  the    equivalent  of    a  quadrantal  version    round 

4/7            I                I 
— —i-\ T=j -\ i^k   followed    by  a  quadrantal  version  round 

2  2^2  2  r 2 

244 

Prob.  49.  In  the  example  on  p.  459  let  b  =  25°  and  c  =  50°;  cal- 
culate out  the  cosine  and  the  directed  sine  of  the  product  angle. 

Prob.  50.  In  the  above  example  calculate  the  cosine  and  the 
directed  sine  up  to  and  inclusive  of  the  fourth  power  of  the  bino- 
mial. (Ans.  cos  =  •9735-) 

Prob.   51.  Calculate   the  first    four   terms   of   the   series  when 

Prob.  52.  From  the  fundamental  theorem  of  spherical  trigo- 
nometry deduce  the  polar  theorem  with  respect  to  both  the  cosine 
and  the  directed  sine. 

Prob.  53.  Prove  that  if  cx'^y  ^*,  y^  denote  the  three  versors  of  a 
spherical  triangle,  then 

sin  /3y  _  sin  ya  _  sin  a/3 
sin  a  sin  b  sin  ^  ' 

Art.  10.    Composition  of  Rotations. 

A  version  refers  to  the  change  of  direction  of  a  line,  but  a 
rotation  refers  to  a  rigid  body.     The  composi-     g 
tion  of  rotations  is  a  different  matter  from  the 
composition  of  versions. 

Effect  of  a  Finite  Rotation  on  a  Line. — Sup- 
pose that  a  rigid  body  rotates  6  radians  round 
the  axis  /3  passing  through  the  point  O,  and  that 
R  is  the  radius-vector  from  O  to  some  particle. 
In  the  diagram  OB  represents  the  axis  /3,  and 
OP  the  vector  R,  Draw  OK  and  OL,  the  rectangular  compo- 
nents of  R, 

/30R  =  (cos  ^  +  sin  ^ .  /S^'^yp 


464 


VECTOR    ANALYSIS    AND    QUATERNIONS.  [ChAP.  IX. 


=  r{cos  0  +  sin  6  .  /5"/*)(cos  /?/o  .  /?  +  sin  /?/o .  ftp/3) 

=  r{cos  /3p.  /5-j-cos  0  sin  /3p  .  ftpj3  +  sin  0  sin  ftp.  /3p\, 
When  cos  ftp  =0,  this  reduces  to 

ft'R  =  cos  OR  +  sin  6Y{ftR). 
The  general  result  may  be  written 

ft^R  =  SftR  .  /5+  cos  e(VftR)f:i  +  sin  6^Vy5i?. 

Note  that  {VftR)ft  is  equal  to  V(VftR)ft  because  S/?i?yS  is 
o,  for  it  involves  two  coincident  directions. 

Example. — Let  ft  =  h'  -\~  mj -\-  nk,  where  P  -\- m^  -\- n^  ^^  I 
and  R  ^  xi -\- yj  -\-  zk  \  then  ^ftR  =  /x  -{-  my  -\-  nz 


Y{ftR)ft 


and 


7nz  —  ny 
I 
i 
YftR  = 


nx  —  Iz     ly  —  7nx 


m 

J 

I    ni 
X  y 
i    j 


Hence  ft^R  =■  {Ix  +  my  +  nz){li  +  mj-^^  nk) 


+  cos^ 


mz—  ny 
I 


nx  —  Iz 
m 

J 


ly  —  mx 
n 
k 


+  sin^ 


/    m 

n 

X  y 

z 

i    j 

k 

To  prove  that  ft^p  coincides  with  the  axis  of  yg-VspVsygVs. 
Take  the  more  general  versor  p^.  Let  OP  represent  the  axis 
ft,  AB  the  versor  yg-*/^  BC  the  versor  p^. 
Then  (AB)(BC)  =  AC  ==  DA,  therefore 
>D  (AB)(BC)(AE):=(DA)(AE)  =  DE.  Now 
DE  has  the  same  angle  as  BC,  but  its  axis 
has  been  rotated  round  P  by  the  angle  b. 
Hence  if  6^  =  7r/2,  the  axis  of  ft-^/^p^/^ft^/^ 
will  coincide  with  y5*p.* 

The  exponential   expression    for 

*  This  theorem  was  discovered  by  Cayley.  It  indicates  that  quaternion 
multiplication  in  the  most  general  sense  has  its  physical  meaning  in  the  compo- 
sition of  rotations. 


Art.  10.]  COMPOSITION  of  rotations.  465 

^-V3pV»y5V3  is  ^-i^P"/'+iTp"/'  +  i-5^'^/^,  vvhich  may  be  expanded 
according  to  the  exponential  theorem,  the  successive  powers 
of  the  trinomial  being  formed  according  to  the  multinomial 
theorem. 

Composition  of  Finite  Rotations  round  Axes  which  Inter- 
sect.— Let  13  and  y  denote  the  two  axes  in  space  round  which 
the  successive  rotations  take  place,  and  let  /J*  denote  the  first 
and  y^  the  second.  Let  /^*  X  y'  denote  the  single  rotation 
which  is  equivalent  to  the  two  given  rotations  appHed  in 
succession  ;  the  sign  X  is  introduced  to  distinguish  from  the 
product  of  versors.  It  has  been  shown  in  the  preceding  para- 
graph that 

and  as  the  result  is  a  line,  the  same  principle  applies  to  the 
subsequent  rotation.     Hence 

y\^^p)  =  ;^-^/2(y5-V2pV2y;5V2)-^V2 

because  the  factors  in  a  product  of  versors  can  be  associated  in 
any  manner.     Hence,  reasoning  backwards, 

Let  m  denote  the  cosine  of  /3^/^y'/^,  namely, 

cos  3/2  cos  c/2  —  sin  d/2  sin  ^/2, 
and  n.  v  their  directed  sine,  namely, 

cos  b/2  sin  c/2.y-\-zos,  c/2  sin  b/2  .  y5— sin  b/2  sin  c/2  sin  ^y  .  Py\ 
then  p^  X  y'^  =  m"^  —  n^  -j-  2mn .  r. 

Observation. — The  expression  (/?*/2^V2y  is  not,  as  might  be 
supposed,  identical  with  /?*;/"•  The  former  reduces  to  the  lat- 
ter only  when  ft  and  y  are  the  same  or 
opposite.  In  the  figure  yS*  is  represented 
by  PQ,  y'  by  QR,  ft'f  by  PR,  ygVa^Va  by 
ST,  and  {ft^/y/^J-  by  SU,  which  is  twice 
ST.  The  cosine  of  SU  differs  from  the 
cosine  of  PR  by  the  term  —(sin  b/2  sin  c/2  sin  /3y)\  It  is 
evident  from  the  figure  that  their  axes  are  also  different. 


466  VECTOK    ANALYSIS    AND    QUATERNIONS.  [Chap.  IX. 

Corollary. — When  b  and  c  are  infinitesimals,  cos  p^Xy'=i, 
and  Sin  ft^  X  y'  =  b.  fS  -{-  c  .y^  which  is  the  parallelogram  rule 
for  the  composition  of  infinitesimal  rotations. 

Prob.  54.  Let  /?  =  307/45°,  ^  =  V3,  and  R  =^  21  -  3/+  4^  ; 
calculate  fi  R. 

Prob.  55.  Let  ft  =  9^/90°,  ^  =  ^/4,  R  =  -  /+  2j-sk; 
calculate  fi^R. 

Prob.  56.   Prove  by  multiplying  out  that  ft- ^/zp^/^/S^/^  =  {ft^p}""^^; 

Prob.  57.  Prove  by  means  of  the  exponential  theorem  that 
y-^P^y^  has  an  angle  by  and  that  its  axis  is  y^ft. 

Prob.  58.   Prove  that  the  cosine   of   (ft^/»y'/^y  differs  from  the 

cosine  of  ft^y*"  by  —  (sin  -  sin  —  sin  ftyj  . 

Prob.  59.  Compare  the  axes  of  {ft^/y/^y  and  ft^y'. 

Prob.  60.  Find  the  value  of  /3^  X  y'  when   /3  =z~^//go°  and 

y  =r^//9o°. 

Prob.  61.  Find  the  single  rotation  equivalent  to  /'^/^  Xy'^/^  X  y^'^A 
Prob.  62.  Prove  that  successive  rotations  about  radii  to  two 
corners  of  a  spherical  triangle  and  through  angles  double  of  those 
of  the  triangle  are  equivalent  to  a  single  rotation  about  the  radius 
to  the  third  corner,  and  through  an  angle  double  of  the  external 
angle  of  the  triangle. 


Art.  1.]  INTRODUCTION.  467 


Chapter  X. 

PROBABILITY  AND   THEORY   OF   ERRORS. 

By  Robert  S.  Woodward, 
Professor  of  Mechanics  in  Columbia  University. 

Art.  1.    Introduction. 

It  is  a  curious  circumstance  that  a  science  so  profoundly 
mathematical  as  the  theory  of  probability  should  have  origi- 
nated in  the  games  of  chance  which  occupy  the  thoughtless 
and  profligate.''^  That  such  is  the  case  is  sufficiently  attested 
by  the  fact  that  much  of  the  terminology  of  the  science  and 
many  of  its  familiar  illustrations  are  drawn  directly  from  the 
vocabulary  and  the  paraphernalia  of  the  gambler  and  the  trick- 
ster. It  is  somewhat  surprising,  also,  considering  the  antiquity 
of  games  of  chance,  that  formal  reasoning  on  the  simpler 
questions  in  probabiHty  did  not  begin  before  the  time  of  Pascal 
and  Fermat.  Pascal  was  led  to  consider  the  subject  during  the 
year  1654  through  a  problem  proposed  to  him  by  the  Chevalier 
de  M6re,  a  reputed  gambler.f  The  problem  in  question  is 
known  as  the  problem  of  points  and  may  be  stated  as  follows : 
two  players  need  each  a  given  number  of  points  to  win  at  a 
certain  stage  of  their  game ;  if  they  stop  at  this  stage,  how  should 
the  stakes  be  divided?  Pascal  corresponded  with  his  friend 
Fermat  on  this  question  ;  and  it  appears  that  the  letters  which 
passed  between  them  contained  the  earliest  distinct  formulation 
of  principles  falling  within  the  theory  of  probability.     These 

*  The  historical  facts  referred  to  in  this  article  are  drawn  mostly  from  Tod- 
hunter's  History  of  the  Mathematical  Theory  of  Probability  from  the  time  of 
Pascal  to  that  of  Laplace  (Cambridge  and  London,  1865). 

f  "  Un  probl^me  relatif  aux  jeux  de  hasard,  propos6  k  un  austere  jans6niste 
par  un  homme  du  monde,  a  6t6  I'origine  du  calcul  des  probabilit6s."  Poisson, 
Recherches  surla  Probability  des  Jugements  (Paris,  1837). 


468  PROBABILITY    AND    THEORY    OF   ERRORS.  [Chap.  X.. 

acute  thinkers,  however,  accomplished  little  more  than  a  correct 
start  in  the  science.  Each  seemed  to  rest  content  at  the  time- 
with  the  approbation  of  the  other.  Pascal  soon  renounced 
such  mundane  studies  altogether ;  Fermat  had  only  the  scant 
leisure  of  a  life  busy  with  affairs  to  devote  to  mathematics; 
and  both  died  soon  after  the  epoch  in  question,— Pascal  in 
1662,  and  Fermat  in  1665. 

A  subject  which  had  attracted  the  attention  of  such  dis- 
tinguished mathematicians  could  not  fail  to  excite  the  interest 
of  their  contemporaries  and  successors.  Amongst  the  former 
Huygens  is  the  most  noted.  He  has  the  honor  of  publishing 
the  first  treatise^  on  the  subject.  It  contains  only  fourteen 
propositions  and  is  devoted  entirely  to  games  of  chance,  but  it 
gave  the  best  account  of  the  theory  down  to  the  beginning  of 
the  eighteenth  century,  when  it  was  superseded  by  the  more  elab- 
orate works  of  James  Bernoulli,f  Montmort,:f  and  De  Moivre.g 
Through  the  labors  of  the  latter  authors  the  mathematical 
theory  of  probabiHty  was  greatly  extended.  They  attacked, 
quite  successfully  in  the  main,  the  most  difificult  problems ; 
and  great  credit  is  due  them  for  the  energy  and  ability  dis- 
played in  developing  a  science  which  seemed  at  the  time  to 
have  no  higher  aim  than  intellectual  diversion.]  Their  names, 
undoubtedly,  with  one  exception,  that  of  Laplace,  are  the  most 
important  in  the  history  of  probability. 

Since  the  beginning  of  the  eighteenth  century  almost  every 
mathematician  of  note  has  been  a  contributor  to  or  an  expos- 
itor of  the  theory  of  probability.  Nicolas,  Daniel,  and  John 
Bernoulli,  Simpson,  Euler,  d'Alembert,  Bayes,  Lagrange,  Lam- 
bert, Condorcet,  and  Laplace  are  the  principal  names  which 
figure  in  the  history  of  the  subject  during  the  hundred  years 

*De  Ratiociniis  in  Ludo  Aleae,  1657. 

f  Ars  Conjectandi,  1713. 

if  Essai  d' Analyse  sur  les  Jeux  de  Hazards,  1708. 

§The  Doctrine  of  Chances,  1718. 

I!  Todhunter  says  of  Montmort,  for  example,  "  In  1708  he  published  his 
work  on  Chances,  where  with  the  courage  of  Columbus  he  revealed  a  new  world. 
to  Mathematicians." 


Art.  1.]  INTRODUCTION.  469 

ending  with  the  first  quarter  of  the  present  century.  Of  the 
contributions  from  this  brilliant  array  of  mathematical  talent, 
the  Th^orie  Analytique  des  Probabilit^s  of  Laplace  is  by  far 
the  most  profound  and  comprehensive.  It  is,  like  his  M^- 
•canique  Celeste  in  dynamical  astronomy,  still  the  most  elabo- 
rate treatise  on  the  subject.  An  idea  of  the  grand  scale  of  the 
work  in  its  present  form*  may  be  gained  by  the  facts  that  the 
non-mathematical  introductionf  covers  about  one  hundred  and 
fifty  quarto  pages ;  and  that,  in  spite  of  the  extraordinary 
brevity  of  mathematical  language,  the  pure  theory  and  its  ac- 
cessories and  applications  require  about  six  hundred  and  fifty 
pages. 

From  the  epoch  of  Laplace  down  to  the  present  time  the 
extensions  of  the  science  have  been  most  noteworthy  in  the 
fields  of  practical  applications,  as  in  the  adjustment  of  obser- 
vations, and  in  problems  of  insurance,  statistics,  etc.  Amongst 
the  most  important  of  the  pioneers  in  these  fields  should 
be  mentioned  Poisson,  Gauss,  Bessel,  and  De  Morgan.  Nu- 
merous authors,  also,  have  done  much  to  simplify  one  or  an- 
other branch  of  the  subject  and  thus  bring  it  within  the  range 
of  elementary  presentation.  The  fundamental  principles  of 
the  theory  are,  indeed,  now  accessible  in  the  best  text-books 
on  algebra :  and  there  are  many  excellent  treatises  on  the  pure 
theory  and  its  various  applications. 

Of  all  the  applications  of  the  doctrine  of  probability  none 
is  of  greater  utility  than  the  theory  of  errors.  In  astronomy, 
geodesy,  physics,  and  chemistry,  as  in  every  science  which  at- 
tains precision  in  measuring,  weighing,  and  computing,  a 
knowledge  of  the  theory  of  errors  is  indispensable.  By  the  aid 
of  this  theory  the  exact  sciences  have  made  great  progress  dur- 

*The  form  of  the  third  edition  published  in  1820,  and  of  Vol.  VII  of  the 
complete  works  of  Laplace  recently  republished  under  the  auspices  of  the 
Academic  des  Sciences  by  Gauthier-Villars.     This  Vol.  VII  bears  the  date  1886. 

f  *'  Cette  Introduction,"  writes  Laplace,  "est  le  developpement  d'une  Le^on 
sur  les  ProbabiHt6s,  que  je  donnai  en  1795,  aux  Ecoles  Normales,  ou  je  fus  ap- 
pel6  comme  professeur  de  Math^matiques  avec  Lagrange,  par  un  d6cret  de  la 
Convention  nationale." 


470  PROBABILITY    AND    THEORY    OF    ERRORS.  [ChAP.  X. 

ing  the  present  century,  not  only  in  the  actual  determination 
of  the  constants  of  nature,  but  also  in  the  fixation  of  clear 
ideas  as  to  the  possibilities  of  future  conquests  in  the  same  di- 
rection. Nothing,  for  example,  is  more  satisfactory  and  in- 
structive in  the  history  of  science  than  the  success  with  which 
the  unique  method  of  least  squares  has  been  appHed  to  the 
problems  presented  by  the  earth  and  the  other  members  of  the 
solar  system.  So  great,  in  fact,  are  the  practical  value  and 
theoretical  importance  of  the  method  of  least  squares,  that  it  is 
frequently  mistaken  for  the  whole  theory  of  errors,  and  is 
sometimes  regarded  as  embodying  the  major  part  of  the  doc- 
trine of  probability  itself. 

As  may  be  inferred  from  this  brief  sketch,  the  theory  of 
probability  and  its  more  important  applications  now  constitute 
an  extensive  body  of  mathematical  principles  and  precepts. 
Obviously,  therefore,  it  will  be  impossible  within  the  limits  of 
a  single  chapter  of  this  volume  to  do  more  than  give  an  out- 
line of  the  sahent  features  of  the  subject.  It  is  hoped,  how- 
ever, in  accordance  with  the  general  plan  of  the  volume,  that 
such  outline  will  prove  suggestive  and  helpful  to  those  who 
may  come  to  the  science  for  the  first  time,  and  also  to  those 
who,  while  somewhat  famiUar  with  the  difftculties  to  be  over- 
tome,  have  not  acquired  a  working  knowledge  of  the  subject. 
Effort  has  been  made  especially  to  clear  up  the  difficulties  of 
the  theory  of  errors  by  presenting  a  somewhat  broader  view  of 
the  elements  of  the  subject  than  is  found  in  the  standard 
treatises,  which  confine  attention  almost  exclusively  to  the 
method  of  least  squares.  This  chapter  stops  short  of  that 
method,  and  seeks  to  supply  those  phases  of  the  theory  which 
are  either  notably  lacking  or  notably  erroneous  in  works 
hitherto  published.  It  is  believed,  also,  that  the  elements  here 
presented  are  essential  to  an  adequate  understanding  of  the 
well-worked  domain  of  least  squares.* 

*The  author  has  given  a  brief  but  comprehensive  statement  of  the  method 
of  least  squares  in  the  volume  of  Geographical  Tables  published  by  the  Smith- 
sonion  Institution,  1894. 


Art.  2.] 


PERMUTATIONS. 


471 


Art.  2.    Permutations. 

The  formulas  and  results  of  the  theory  of  permutations 
and  combinations  are  often  needed  for  the  statement  and  so- 
lution of  problems  in  probabilities.  This  theory  is  now  to  be 
found  in  most  works  on  algebra,  and  it  will  therefore  suffice 
here  to  state  the  principal  formulas  and  illustrate  their  mean- 
ing by  a  few  numerical  examples. 

The  number  of  permutations  of  n  things  taken  r  in  a  group 
is  expressed  by  the  formula 

(7i\  =  n{n  -  i){n  _  2)  ...(;/-  r  +  i).  (i) 

Thus,  to  illustrate,  the  number  of  ways  the  four  letters  «,  by 
Cy  d  can  be  arranged  in  groups  of  two  is  4 .  3  =  1 2,  and  the  groups 


are 


bdy     db,     cdy     dc. 


ab,     ba,     ac,     ca,     ad,     da,     bc^     cb^ 
Similarly,  the  formula  gives  for 
n=    I  and  r  =  2,         (3),  =  3-2 
«  =    7    -     r=3,         (7)3  ==7.6.  5. 
«  =  10    "     r  =  6,       (10)5—10.9.8 

The  results  which  follow  from   equation  (i)  when  «  and  r 
do  riot  exceed  10  each  are  embodied  in  the  following  table  : 
Values  of  Permutations. 


=  6, 
=  210, 
7.6.5  =  151 200. 


zo 

9 

8 

7 

6 

5 

4 

3 

3 

2 

2 

I 
I 

X 

10 

9 

8 

7 

6 

5 

4 

2 

90 

72 

56 

42 

30 

20 

12 

6 

2 

3 

720 

504 

336 

210 

120 

60 

24 

6 

4 

5040 

3024 

1680 

840 

360 

120 

24 

1 

30240 

15120 

6720 

2520 

720 

120 

I5I200 

60480 

20160 

5040 

720 

7 

604800 

181440 

40320 

5040 

8 

18:4400 

362880 

40320 

9 

3628800 

362880 

10 

3628800 

4 

I 

Sp 

9864100 

986409 

109600 

13699 

1956 

325 

64 

15 

The  use  of  this  table  is  obvious.  Thus,  the  number  of  per- 
mutations of  eight  things  in  groups  of  five  each  is  found  in  the 
fifth  line  of  the  column  headed  with  the  number  8.     It  will  be 


472  PROBABILITY    AND    THEORY    OF    ERRORS.  [Chap.  X. 

noticed  that  the  last  two  numbers  in  each  column  (excepting 
that  headed  with  i)  are  the  same.  This  accords  with  the  for- 
mula, which  gives  for  the  number  of  permutations  of  n  things 
in  groups  of  n  the  same  value  as  for  n  things  in  groups  of 
(«  —  i).  It  will  also  be  remarked  that  the  last  number  in  each 
column  of  the  table  is  the  factorial,  n\^  of  the  number  n  at  the 
head  of  the  column.  For  example,  in  the  column  under  7,  the 
last  number  is  5040  =1.2.3.4.5.6.7  =  7!. 

The  total  number  of  permutations  of  n  things  takea  singly, 
in  groups  of  two,  three,  etc.,  is  found  by  summing  the  numbers 
given  by  equation  (i)  for  all  values  of  r  from  i  to  n.  Calling 
this  total  or  sum  5^,  it  will  be  given  by 

Sp  =  ^«.  (2) 

To  illustrate,  suppose  n  =  ^,  and,  to  fix  the  ideas,  let  the 
three  things  be  the  three  digits  i,  2,  3.  Then  from  the  above 
table  it  is  seen  that  5^  =  3  +  6  -[-  6  =  15  ;  or,  that  the  number 
of  numbers  (all  different)  which  can  be  formed  from  those  dig- 
its is  fifteen.  These  numbers  are  I,  2,  3  ;  12,  13,  21,  23,  31,  32  ; 
123,  132,  213,  231,  312,  321. 

The  values  of  Sp  (or  n  =  1,2,...  10  are  given  under  the 
corresponding  columns  of  the  above  table.  But  when  7i  is 
large  the  direct  summation  indicated  by  (2)  is  tedious,  if  not 
impracticable.  Hence  a  more  convenient  formula  is  desirable. 
To  get  this,  observe  that  (i)  may  be  written 

if  r  is  restricted  to  integer  values  between  i  and  (n  —  i),  both 
inclusive.  This  suffices  to  give  all  terms  which  appear  in  the 
right-hand  member  of  (2),  since  the  number  of  permutations 
for  r  =  (;^  —  i)  is  the  same  as  for  r  =  n.  Hence  it  appears 
that 

^  '    I     '    I  .  2   '  (n  —  i)\ 


.Art.  3.]  COMBINATIONS.  473 

But  as  n  increases,  the  series  by  which  n !  is  here  multiplied 
approximates  rapidly  towards  the  base  of  natural  logarithms'; 
that  is,  towards 

^  =  27182818 +>         log  ^  =  0.4342945. 
Hence  for  large  values  of  n 

Sp  =  nle,  approximately.*  (3) 

To  get  an  idea  of  the  degree  of  approximation  of  (3),  sup- 
pose n  =  g.     Then  the   computation  runs  thus  (see  values  in 

the  above  table) : 

log 
9!  =  362880         5-5597630 

e  04342945 

91^  =  986410         5-9940575 

Sp  =  986409  by  equation  (2). 

The  error  in  this  case  is  thus  seen  to  be  only  one  unit,  or 
about  one-miUionth  of  5^.f 

Prob.  I.  Tabulate  a  list  of  the  numbers  of  three  figures  each 
which  can  be  formed  from  the  first  five  digits  i,  .  .  .  5.  How  many 
numbers  can  be  formed  from  the  nine  digits  ? 

Prob.  2.  Is  Sfi  always  an  odd  number  for  n  odd  ?  Observe 
values  of  Sp  in  the  table  above. 

Art.  3.    Combinations. 

In  permutations  attention  is  given  to  the  order  of  arrange- 
ment of  the  things  considered.  In  combinations  no  regard  is 
paid  to  the  order  of  arrangement.  Thus,  the  permutations  of 
the  letters  a,  b,  c,  d  in  groups  of  three  are 

(abc)  {abd)  bac  bad  acb  {acd)  cab  cad 
adb  adc  dab  dac  bca  {bed)  cba  cbd 
bda       bdc       dba     dbc     cda      cdb       dca     deb 

*  See  Art.  6  for  a  formula  for  computing  n\  when  «  is  a  large  number, 

t  When  large  numbers  are  to  be  dealt  with,  equations  (i)'  and  (3)  are  easily 

managed  by  logarithms,  especially  if  a  table  of  values  of  log  («!)  is  available. 

Such  tables  are  given  to  six  places  in  De  Morgan's  treatise  on  Probability  in 

the    Encyclopaedia    Metropolitana,   and   to   five  places   in   Shortrede's   Tables 

(Vol.  I,  1849). 


474  PROBABILITY    AND    THEORY    OF    ERRORS.  [Chap.  X. 

But  if  the  order  of  arrangement  is  ignored  all  of  these  are 
seen  to  be  repetitions  of  the  groups  enclosed  in  parentheses, 
namely,  {abc\  {abd),  {acd),  {bed).  Hence  in  this  case  out  of 
twenty-four  permutations  there  are  only  four  combinations. 

A  general  formula  for  computing  the  number  of  combina- 
tions of  n  things  taken  in  groups  of  r  things  is  easily  derived. 
For  the  number  of  permutations  of  n  things  in  groups  of  r  is 
by  (i)  of  Art.  2 

(n)^  =  n{fi  —  i){n  —  2)  ,  .  .  {n  —  r  -{-  i)\ 

and  since  each  group  of  r  things  gives  1.2.3.  ..r  =  r!  Per- 
mutations, the  number  of  combinations  must  be  the  quotient 
of  {n)^  by  r!.  Denote  this  number  by  C{n)r,  Then  the  gen- 
eral formula  is 

C(«).  =  ^("-^)(n-2y..(n-r+i)  ^^^ 

This  formula  gives,  for  example,  in  the  case  of  the  four  let- 
ters a,  bf  Cf  d  taken  in  groups  of  three,  as  considered  above, 

^^'"^  1.2.3  ^ 

Multiply  both  numerator  and  denominator  of  the  right-hand 
member  of  (i)  by  (n  —  r)\     The  result  is 

C{n)r  =  -77 — '-^rv  W 

which  shows  that  the  number  of  combinations  of  n  things  in 
groups  of  r  is  the  same  as  the  number  of  combinations  of  9t 
things  in  groups  of  {n  —  r).  Thus,  the  number  of  combina- 
tions of  the  first  ten  letters  a,  b,  c  ,  ,  .y  in  groups  of  three  or 
seven  is 

10! 

-1 :  =   120. 

3 '.  7  '• 
The  following  table  gives  the  values  C{n)^  for  all  values  of 
n  and  r  from  i  to  10. 

The  mode  of  using  this  table  is  evident.  For  example,  the 
number  of  combinations  of  eight  things  in  sets  of  five  each  is 
found  on  the  fifth  line  of  the  column  headed  8  to  be  56. 


Art.  3.] 


COMBINATIONS. 

Values  of  Combinations. 


475 


10 

9 

8 

7 

6 

6 

5 

5 

4 

4 

3 

3 

2 
2 

I 

I 

I 

lO 

9 

8 

7 

2 

45 

36 

28 

21 

15 

10 

6 

3 

I 

3 

I20 

84 

56 

35 

20 

10 

4 

I 

4 

2IO 

126 

70 

35 

15 

5 

I 

1 

252 

126 

56 

21 

6 

I 

210 

84 

28 

7 

I 

7 

120 

36 

8 

I 

8 

45 

9 

I 

9 

10 

I 

10 

I 

•5 

7 

3 

Sc 

1023 

5" 

255 

127 

63 

31 

It  will  be  observed  that  the  numbers  in  any  column  show 
a  maximum  value  when  n  is  even  and  two  equal  maximum 
values  when  n  is  odd.  That  this  should  be  so  is  easily  seen 
from  (i)',  which  shows  that  C(n)^  will  be  a  maximum  for  any 
value  of  Jt  when  r\  {ft  —  r)\  is  a.  minimum.  For  n  even  this  is 
a  minimum  for  r  =  ^n;  while  for  n  odd  it  has  equal  minimum 
values  for  r  =  ^{n  —  i)  and  r  =  j^{n  -\-  i).     Thus, 


ri    i 

maximum  of  C{n)r  =  -, — ^  for  n  even, 


g'j 


(2> 


n  -\-  I    ?i 


for  n  odd. 


The  total  number  of  combinations  of  iz  things  taken  singly,. 
in  groups  of  two,  three,  etc.,  is  found  by  summing  the  numbers 
given  by  (i)  for  all  values  of  r  from  i  to  n  both  inclusive. 
Calling  this  total  or  sum  S^y 

The  same  sum  will  also  come  from  (i)'by  giving  to  rail  values 
from  I  to  {n  —  i),  both  inclusive,  summing  the  results,  and  in- 
creasing their  aggregate  by  unity.     Thus  by  either  process 


S,  =  n  + 


n{n  —  i)       n{n  —  \){n  —3) 


I  .  2 


+ 


1.2.3 


+  ... +  ;.+  !. 


476  PROBABILITY    AND    THEORY    OF    ERRORS.  [Chap.  X. 

The   second  member   of   this  equation  is  evidently  equal   to 

(i  +  0** "~  ^'     Hence 

S,=  2C{n),  =  2^-1,  (3) 

The  values  of  S^  for  values  of  ;/  and  r  from  i  to  lo  are  given 
tunder  the  corresponding  columns  of  the  above  table. 

Prob.  3.  How  many  different  squads  of  ten  men  each  can  be 
■formed  from  a  company  of  100  men  ? 

Prob.  4  How  many  triangles  are  formed  by  six  straight  lines 
each  of  which  intersects  the  other  five  ? 

Prob.  5.  Examine  this  statement  :  "  In  dealing  a  pack  of  cards 
the  number  of  hands,  of  thirteen  cards  each,  which  can  be  produced 
is  635  013  559  600.  But  in  whist  four  hands  are  simultaneously  held, 
and  the  number  of  distinct  deals  .  .  .  would  require  twenty-eight 
figures  to  express  it."  * 

Prob.  6.  Assuming  combination  always  possible,  and  disregarding 
the  question  of  proportions,  find  how  many  different  substances 
could  be  produced  by  combining  the  seventy-three  chemical  ele- 
ments.    . 

Art.  4.     Direct  Probabilities. 

If  it  is  known  that  one  of  two  events  must  occur  in  any 
trial  or  instance,  and  that  the  first  can  occur  in  a  ways  and  the 
second  in  d  ways,  all  of  which  ways  are  equally  likely  to  hap- 
pen, then  the  probability  that  the  first  will  happen  is  expressed 
mathematically  by  the  fraction  a/{a-\-d),  while  the  probability 
that  the  second  will  happen  is  d/(a  -|-  ^).  Such  events  are  said 
to  be  mutually  exclusive.  Denote  their  probabilities  by  p  and 
g  respectively.     Then  there  result 

the  last  equation  following  from  the  first  two  and  being  the 
mathematical  expression  for  the  certainty  that  one  of  the  two 
events  must  happen. 

Thus,  to  illustrate,  in  tossing  a  coin  it  must  give  "  head  "  or 
■*'  tail"  ',  a  :=  d  =  ly  and  p  =  g  =  1/2.  Again,  if  an  urn  contain 
a  =  S  white  and  d  =  S  black  balls,  the  probability  of  drawing 

*  Jevons,  Principles  of  Science,  p.  217. 


Art.  4.]  DIRECT  probabilities.  477' 

a  white  ball  in  one  trial  is  /  =  5/13  and  that  of  drawing  a 
black  one  q  =  8/13. 

Similarly,  if  there  are  several  mutually  exclusive  events 
which  can  occur  in  ay  d,  c .  .  ,  ways  respectively,  their  probabil-^ 
ities  /,  ^,  r  .  .  .    are  given  by 

a  ^  _  ^ 

^  =  a-\-d-{-c-{-,  .' '  ^  ^  a  +  J+T+TT, '  ^  "^  a-\-b-\-c-\-.,. '' 

For  example,  if  an  urn  contain  a=:  4.  white,  ^  =  5  black,, 
and  c  :=  6  red  balls,  the  probabilities  of  drawing  a  white,  black, 
and  red  ball  at  a  single  trial  are  />  =  4/i5,  ^=5/15,  and 
r  =  6/15,  respectively. 

Formulas  (i)  and  (2)  may  be  applied  to  a  wide  variety  of 
cases,  but  it  must  suffice  here  to  give  only  a  few  such.  As  a 
first  illustration,  consider  the  probability  of  drawing  at  random 
a  number  of  three  figures  from  the  entire  list  of  numbers  which 
can  be  formed  from  the  first  seven  digits.  A  glance  at  the 
table  of  Art.  i  shows  that  the  symbols  of  formula  (i)  have  in 
this  case  the  values  a  =  210,  and  a-\-d=i^6gg.  Hence 
fi  =  13489,  and  /  =  210/13699  ;  that  is,  the  probabiHty  in  ques- 
tion is  about  1/65. 

Secondly,  what  is  the  probability  of  holding  in  a  hand  of 

whist  all  the  cards  of  one  suit  ?     Formula  (i)  of  Art.  3  shows 

'  that  the  number  of  different  hands  of  thirteen  cards  each  which 

may  be  formed  from  a  pack  of  fifty-two  cards  is 

52.51.  50...  40 

^ — ^ — ^ !L_^63Coi3  559600, 

I  .  2  .  3  ...  13  -^-^      -^  -'•'^        ' 

and  the  probability  required  is  the  reciprocal  of  this  number. 
The  probability  against  this  event  is,  therefore,  very  nearly 
unity. 

Thirdly,  consider  the  probabilities  presented  by  the  case  of 
an  urn  containing  4  white,  5  black,  and  6  red  balls,  from  which 
at  a  single  trial  three  balls  are  to  be  drawn.  Evidently  the 
triad  of  balls  drawn  may  be  all  white,  all  black,  all  red,  partly 
white  and  black,  partly  white  and  red,  partly  black  and  red,  or 


478  PROBABILITY    AND    THEORY    OF    ERRORS.  [Chap.  X. 

one  each  of  the  white,  black,  and  red.  There  are  thus  seven 
different  probabiUties  to  be  taken  into  account.  The  theory 
of  combinations  shows  (see  equation  (i).  Art.  3)  that  the  total 

number  of 

4.3.2 
White  triads  =   =     4  =  a 


Black  triads  =   ^^eT"  =   10  =  b 

6.5.4 
Red  triads  =   — -? =   20  =  ^ 


I  , 

.  2  . 

3 

5 

.4. 

3 

6 

6 

.5. 

4 

6 

9 

.8. 

7 

6 

10 

•9 

.8 

6 

II 

.10.9 

6 

4. 

■5- 

6 

White  and  black  triads         =   ^ —  (  4+10)=  'jo  =  d 

10.  9 . 8 
White  and  red  triads  =  — ^ —    —  (  4+20)=  96  =  ^ 

Black  and  red  triads  =  — ^— _ L^_(io-|-2o)  =  i35  =/ 

White,  black,  and  red  triads  =  4.5.6  — 120  =  ^ 

Sum  =  455 
The  total  number  of  these  triads  is  455,  and  is,  as  it  should 
be,  the  number  of  combinations  in  groups  of  three  each  of  the 
whole  number  of  balls.  Hence  formulas  (2)  give  the  seven 
different  probabilities  which  follow,  using  the  initial  letters 
w,  by  r  to  indicate  the  colors  represented  in  a  triad  : 


For  a  triad  www 

/  =      4/455, 

u     u      u       ^^^ 

g=    10/455, 

a      a       u         ^^^ 

r=    20/455, 

it    u     It    :  yjiju^  or  wbb 

s=    70/455, 

u    u     «      wwr  or  wrr 

t=    96/455, 

"    *'     "       bbr  or  brr 

^=  135/455, 

a     u      u       ^^^ 

v=  120/455. 

Prob.  7.  When  three  dice  are  thrown  together,  what  is  the  prob- 
ability that  the  throw  will  be  greater  than  9  ? 

Prob.  8.  Write  down  a  literal  formula  for  the  probabilities  of  the 
several  possible  triads  considered  in  the  above  question  of  the  balls, 
supposing  the  numbers  of  white,  black,  and  red  balls  to  be  /,  m,  n, 
respectively. 


Art.  5.]  probability  of  concurrent  events.  479 

Art.  5.    Probability  of  Concurrent  Events. 

If  the  probabilities  of  two  independent  events  are/^  and 
/j,  respectively,  the  probability  of  their  concurrence  in  any 
single  instance  is  p^p^.  Thus,  suppose  there  are  two  urns 
U^  and  U^,  the  first  of  which  contains  a^  white  and  b^  black 
balls,  and  the  second  a^  white  and  b^  black  balls.  Then  the 
probability  of  drawing  a  white  ball  from  U^  is  p^  ==  ^i/(^i  +  b^y 
while  that  of  drawing  a  white  ball  from  U^  is/^  =  ^aA'^aH-  b^. 
The  total  number  of  different  pairs  of  balls  which  can  be  formed 
from  the  entire  number  of  balls  is  {a^  +  b^ia^  -f"  ^a)*  Of  these 
pairs  a^a^  are  favorable  to  the  concurrence  of  white  in  simul- 
taneous or  successive  drawings  from  the  two  urns.  Hence  the 
probability  of  a  concurrence  of 

white  with  white  =  a^aj{a^  -\-  b^ia^  +  b^, 

white  with  black  =  {ajb^  +  ^J^^/^^x  +  b^(a^  +  b^, 

black  with  black  =  bjbj{a,  +  b^{a^  +  b^, 

and  the  sum  of  these  is  unity,  as  required  by  equations  (2)  of 
Article  4. 

In  general,  if /j,  /,,  A  •  •  •  denote  the  probabilities  of  several 
independent  events,  and  P  denote  the  probability  of  their 
concurrence, 

P-=^PJ.P.'  (I) 

To  illustrate  this  formula,  suppose  there    is  required  the 
probability  of  getting  three  aces  with  three  dice  thrown  simul- 
taneously.    In  this  case/i  =p^=p^=  1/6  and 
P={i/6y=  1/2 16. 

Similarly,  if  two  dice  are  thrown  simultaneously  the  proba- 
bility that  the  sum  of  the  numbers  shown  will  be  11  is  2/36; 
and  the  probability  that  this  sum  1 1  will  appear  in  two  succes- 
sive throws  of  the  same  pair  of  dice  is  4/36.36. 

The  probability  that  the  alternatives  of  a  series  of  events 
will  concur  is  evidently  given  by 

Q  =  q.g.q^ . . .  =  (i  - p,){i  -p:){i  -a)  (2) 

Thus,  in  the  case  of  the  three  dice  mentioned  above,  the 
probability  that  each  will  show  something  other  than  an  ace  is 


480  PROBABILITY    AND    THEORY    OF    ERRORS.  [ChAP.  X. 

q^=i  q^=  q^  =  5/6,  and  the  probability  that  they  will  concur  in 
this  \s  Q  =  125/216. 

Many  cases  of  interest  occur  for  the  application  of  (i)  and 
(2).  One  of  the  most  important  of  these  is  furnished  by  suc- 
cessive trials  of  the  same  event.  Consider,  for  example,  what 
may  happen  in  n  trials  of  an  event  for  which  the  probability 
is  p  and  against  which  the  probability  is  g.  The  probability 
that  the  event  will  occur  every  time  is/**.  The  probability  that 
the  event  will  occur  {n  —  i)  times  in  succession  and  then  fail  is 
/"  ■  ^g.  But  if  the  order  of  occurrence  is  disregarded  this  last 
combination  may  arrive  in  n  different  ways  ;  so  that  the  prob- 
ability that  the  event  will  occur  {n  —  i)  times  and  fail  once  is 
7ip^~''g.  Similarly,  the  probability  that  the  event  will  happen 
{n  —  2)  times  and  fail  twice  is  \n{ri  —  i)/""y ;  etc.  That  is, 
the  probabilities  of  the  several  possible  occurrences  are  given  by 
the  corresponding  terms  in  the  development  of  (/  -j-  g^. 

By  the  same  reasoning  used  to  get  equations  (2)  of  Art. 
3  it  may  be  shown  that  the  maximum  term  in  the  expansion 
of  (/  -|-  gY  is  that  in  which  the  exponent  m,  say,  of  g  is 
the  whole  number  lying  between  {n-{-  i)g  —  i  and  (n -{-  i)g. 
In  other  words,  the  most  probable  result  in  n  trials  is  the 
occurrence  of  the  event  {n  —  m)  times  and  its  failure  m 
times.  When  7t  is  large  this  means  that  the  most  probable  of 
all  possible  results  is  that  in  which  the  event  occurs  n  —  nq 
=  n{\  —  g)—  np  times  and  fails  ^^tinrtes.  Thus,  if  the  event 
be  that  of  throwing  an  ace  with  a  single  die  the  most  probable 
of  the  possible  results  in  600  throws  is  that  of  100  aces  and 
500  failures. 

Since  q""  is  the  probability  that  the  event  will  fail  every  time 
in  n  trials,  the  probability  that  it  will  occur  at  least  once  in  n 
trials  is  I  —  ^^    ^Calling  this  probability  r,* 

r=^-qn^^  -{i-py,  (3) 

If  r  in  this  equation  be  replaced  by  1/2,  the  corresponding^ 
value    of    n  is  the  number   of   trials   essential  to  render  the 

*  See  Poisson's  Probability  des  Jugements,  pp.  40,  41. 


Art.  5.]  probability  of  concurrent  events.  481 

chances  even  that  the  event  whose  probabih'ty  \s  p  will  occur 
at  least  once.     Thus,  in  this  case,  the  value  of  n  is  given  by 

log  2 

n—  —  , ^^ ^,. 

log  (I-/) 

This  shows,  for  example,  if  the  event  be  the  throwing  of  double 
sixes  with  two  dice,  for  which  p  =  1/36,  that  the  chances  are 
even  (r  =  1/2)  that  in  25  throws  {n  —  24.614  by  the  formula) 
double  sixes  will  appear  at  least  once. 

Equation  (3)  shows  that  however  small/  may  be,  so  long  as 
it  is  finite,  n  may  be  taken  so  large  as  to  make  r  approach  in- 
definitely near  to  unity  ;  that  is,  n  may  be  so  large  as  to  render 
it  practically  certain  that  the  event  will  occur  at  least  once. 

When  n  is  large 

/        .\«  .  nin—i)   :     «(«— i)(«— 2),    . 

(I  -pr  =i-np  +  \^f ^^3^3      >'  +  . . . 

-T       ....    ('^^)'        (^^)'     . 


1.2  1.2.3' 

=  e~*^  approximately. 
Thus  an  approximate  value  of  r  is 

r  =  I  -  ^  -  «^         log  ^  =  0.4342495.  (4) 

This  formula  gives,  for  example,  for  the  probability  of  drawing 
the  ace  of  spades  from  a  pack  of  fifty-two  cards  at  least  once  in 
104  trials  r  =  I  —  ^~'  =  0.865,  while  the  exact  formula  (3) 
gives  0.867. 

Similarly,  the  probability  of  the  occurrence  of  the  event  at 
least  /  times  in  n  trials  will  be  given  by  the  sum  of  the  terms 
of  {p  +  qY  from  /»"  up  to  that  in  fq^-^  inclusive.  This  proba- 
bility must  be  carefully  distinguished  from  the  probability  that 
the  event  will  occur  /  times  only  in  the  n  trials,  the  latter  being 
expressed  by  the  single  term  in  p^q""'*-. 

Prob.  9.  Compare  the  probability  of  holding  exactly  four  aces  in 
five  hands  of  whist  with  the  probability  of  -lolding  at  least  four  aces 
in  the  same  number  of  hands. 

Prob.  10.  What  is  the  probability  of  an  event  if  the  chances  are 
even  that  it'occurs  at  least  once  in  a  million  trials  ?  See  equation  (4). 


482  PROBABILITY    AND    THEORY    OF    ERRORS.  [ChaP.  X. 

Art.  6.    Bernoulli's  Theorem. 

Denote  the  exponents  of/  and  q  in  the  maximum  term  of 
(^  _i_  qY  by  yu  and  m  respectively,  and  denote  this  term  by  T. 
Then 

As  shown  in  Art.  5,  yw  in  this  formula  is  the  greatest  whole 
number  in  {n  +  i)/,  and  m  the  greatest  whole  number  in 
(^  _|_  i)^ ;  so  that  when  n  is  large,  yu  and  m  are  sensibly  equal 
to  np  and  ;/^  respectively. 

The  direct  calculation  of  T  by  (i)  is  impracticable  when  n 
is  large.  To  overcome  this  difficulty  the  following  expression 
is  used :  * 

;2!  =  ;2«^-«i/i^(l4-  — +-^+.  .  .).  (2) 

log  ^  =  0.4342495,      log  271  =  0.7981799. 

This  expression  approaches  wV""  4^2 ;r«  as  a  limit  with  the 
increase  of  «,  and  in  this  approximate  form  is  known  as  Stir- 
ling's theorem.  Although  a  rude  approximation  to  n\  for 
small  values  of  n  this  theorem  suffices  in  nearly  all  cases 
wherein  such  probabilities  as  7"  are  desired.  Making  use  of 
the  theorem  in  (i)  it  becomes 

T^     .1 (3) 

V27tnpq 

That  this  formula  affords  a  fair  approximation  even  when 
n  is  small  is  seen  from  the  case  of  a  die  thrown  12  times.  The 
probability  that  any  particular  face  will  appear  in  one  throw  is 
p  —  1/6,  whence  q  =  5/6;  and  the  most  probable  result  in  12 
throws  is  that  in  which  the  particular  face  appears  twice  and 
fails  to  appear  ten  times.  The  probabiHty  of  this  result  com- 
puted from  (3)  is  0.309,  while  the  exact  formula  (i)  gives  0.296. 

The  probability  that  the  event  will  occur  a  number  of  times 

*  This  expression  is  due  to  Laplace,  Th6orie  Analytique  des  Probabilit6s. 
See  also  De  Morgan's  Calculus,  pp.  600-604. 


Art.  6.]  Bernoulli's  theorem.  483 

comprised  between  {^  —  a)  and  (/(  -|-  a)  in  ?i  trials  is  evidently 
expressed  by  the  sum  of  the  terms  in  (/  -j-  gy  for  which  the 
exponent  of  /  has  the  specified  range  of  values.  Calling  this 
probability  R,  putting 

jLi  =  ?i/>  -{-  u,     and     m  =  nq  —  u^ 

and  using  Stirling's  theorem  (which  imphes  that  «  is  a  large 
number),* 

V27t7ipq^         npl  \         nql 

very  nearly ;  and  the  summation  is  with  respect  to  u  from 
u^=^  —  a\.ou^=-\-a.  But  expansion  shows  that  the  natural 
logarithm  of  the  product  of  the  two  binomial  factors  in  this 
equation  is  approximately  —  u'/2npq.     Hence 

R  =    2 ^  ,p-ui/%npq  . 

V27Tnpq 

and,  since  n  is  supposed  large,  this  may  be  replaced  by  a  definite 
integral,  putting 


Thus 


dz  =  l/V2npq,     and     z^  =  u^/2npq, 

+  a/  ^%npq  a/  ^%npq 


R=z—^       e-^'dz  =  -—    \  e-^'dz.  (4) 

S/nJ_  V7tJ 

-a/\^inpq  ^ 

This  equation  expresses  the  theorem  of  James  Bernoulli, 
given  in  his  Ars  Conjectandi,  published  in  171 3. 

The  value  of  the  right-hand  member  of  (4)  varies,  as  it 
should,  between  o  and  i,  and  approaches  the  latter  limit  rap- 
idly as  z  increases.     Thus,  writing  for  brevity 


z 


e-  '^dzy 


*  See  Bertrand,  Calcul  des  ProbabiHt6s,  Paris,  1889,  for  an  extended  discus- 
sion of  the  questions  considered  in  this  Article. 


484  PROBABILITY    AND    THEORY    OF    ERRORS.  [Chap.  X. 

the  following  table  shows  the  march  of  the  integral : 


0 

/ 

z 

I 

z 

/ 

o.oo 

0.000 

0.75 

O.7II 

1.50 

0.966 

.25 

.276 

1. 00 

.843 

1-75 

.987 

.50 

.520 

1.25 

•923 

2.00 

.995 

To  illustrate  the  use  of  (4),  suppose  there  is  required  the 
probabiHty  that  in  60CX)  throws  of  a  die  the  ace  will  appear  a 
number  of  times  which  shall  be  greater  than  1/6  X  6000  —  10 
and  less  than  1/6  X  6000  +  10,  or  a  number  of  times  lying 
between  990  and  10 10.  In  this  case  «r  =  10,  n  —  6000,  /  =  1/6, 
g  =  5/6.  Thus,  a/^2npq  =  10/4^2  .  6000  .  1/6  .  5/6  =  O.245. 
Hence,  by  (4)  and  the  table,  R  =  O.27. 

Prob.  II.  If  the  ratio  of  males  to  females  at  birth  is  105  to  100, 
what  is  the  probability  that  in  the  next  10,000  births  the  number  of 
males  will  fall  within  two  per  cent  of  the  most  probable  number  ? 

Prob.  12.  If  the  chance  is  even  for  head  and  tail  in  tossing  a 
coin,  what  is  the  probability  that  in  a  million  throws  the  difference 
between  heads  and  tails  will  exceed  1500? 

Art.  7.     Inverse  Probabilities.* 

If  an  observed  event  can  be  attributed  to  any  one  of  several 
causes,  what  is  the  probability  that  any  particular  one  of  these 
causes  produced  the  event  ?  To  put  the  question  in  a  concrete 
form,  suppose  a  white  ball  has  been  drawn  from  one  of  twa 
urns,  U^  containing  3  white  and  5  black  balls,  and  £/,  contain- 
ing 2  white  and  4  black  balls  ;  and  that  the  probability  in  favor 
of  each  urn  is  required.  If  U,  is  as  likely  to  have  been 
chosen  as  U^,  the  probability  that  £/,  was  chosen  is  1/2.  After 
such  choice  the  probability  of  drawing  a  white  ball  from  U^  is 
3/8.  Before  drawing,  therefore,  the  probability  of  getting  a 
white  ball  from  U^  was  1/2  X  3/8  =  3/16,  by  Art.  5.  Similarly, 
before  drawing  the  probability  of  getting  a  white  ball  from  U^ 
was  1/2  X  2/6  =  1/6.  These  probabilities  will  remain  un- 
changed if  the  number  of  balls  in  either  urn  be  increased  or 

*  See  Poisson,  Probability  des  Jugements,  pp.  81-83.  , 


Art.  7.]  INVERSE    PROBABILITIES.  465 

diminished  so  long  as  the  ratio  of  white  to  black  balls  is  kept 
constant.  Make  these  numbers  the  same  for  the  two  uins. 
Thus  let  the  first  contain  9  white  and  15  black,  and  the  second 
8  white  and  16  black;  whence  the  above  probabilities  may  be 
written  1/2  x  9/24  and  1/2  X  8/24.  It  is  now  seen  that  there 
are  (9  +  8)  cases  favorable  to  the  production  of  a  white  ball, 
each  of  which  has  the  same  antecedent  probability,  namely,  1/2. 
Since  the  fact  that  a  white  ball  was  drawn  excludes  considera- 
tion of  the  black  balls,  the  probability  that  the  white  ball  came 
from  U^  is  9/17  and  that  it  came  from  U^  is  8/17;  and  the  sum 
of  these  is  unity,  as  it  should  be. 

To  generalize  this  result,  let  there  be  m  causes,  C^,C^,...  C^. 
Denote  their  direct  probabilities  hyq^,  ^,, . . .  q„,\  their  antecedent 
probabilities  by  r,,  r,,  .  .  .  r,„  ;  and  their  resultant  probabilities 
on    the   supposition   of   separate    existence    by  A»Aj  •••/»«• 

That  is, 

/i  =  ^x^i»    A  =  ^.^'  .  .  .  /^  =  q^nrn.'  (I) 

Let  D  be  the  common  denominator  of  the  right-hand  mem- 
bers in  (i),  and  denote  the  corresponding  numerators  of  the 
several  fractions  by  s.,  s^y  ,  .  .  s,^.     Then 

p,  =  sJD,    A  =  -^y^^  ,  .  .p^=  sJD ; 
and  it  is  seen  that  there  are  in  all  (^,  +  ^a  +  •  •  •  ^n)  equally 
possible  cases,  and  that  of  these  s^  are  favorable  to  C^,  s^  to 
(7,,  .  .  .     Hence,  if  /\,  /!,,...  P^  denote  the  probabilities  of 
the  several  causes  on  the  supposition  of  their  coexistence, 

P^  =  V(^.  +S,+  ...Srn)  =/y(A  +  A  +  .  .  .  PJ' 

Thus  in  general 

P,=p,/2/>,     P,  =  pJSp,...P„=p„/2p.  (2) 

To  illustrate  the  meaning  of  these  formulas  by  the  above 
concrete  case  of  the  urns  it  suffices  to  observe  that 
for  f/;,      q^  =  3/8     and     r,  =  1/2, 
for  U^,      q,  =  1/3     and     r,  =  1/2  ; 

whence  A  =  3/i6,    A  =  lA    A  +  A  =  ^7/4^  ; 

and  P,  =  9/17,    P.^  =  8/17. 

As  a  second  illustration,  suppose  it  is  known  that  a  white 


486  PROBABILITY    AND    THEORY    OF    ERRORS.  [ChAP.  X> 

ball  has  been  drawn  from  an  urn  which  originally  contained  m 
balls,  some  of  them  being  black,  if  all  are  not  white.  What  is 
the  probabiHty  that  the  urn  contained  exactly  n  white  balls? 
The  facts  are  consistent  with  m  different  and  equally  probable 
hypotheses  (or  causes),  namely,  that  there  were  i  white  and 
(^fyi  —  i^  black  balls,  2  white  and  {jn  —  2)  black  balls,  etc. 
Hence  in  (i),  q^  =  q^  —  ,  .  ,  —  \,  and 

p^  =  i/niy    A  =  2/m,  .  .  .  pn  =  n/m,  .../,„=  m/m. 
Thus  2p=(i/2){m+i), 

2n 

This  shows,  as  it  evidently  should,  that  n  =  mis  the  most 
probable  number  of  white  balls  in  the  urn.  The  probabiHty 
for  this  number  is  P^  =  2/{m  +  i),  which  reduces,  as  it  ought, 
to  I  for  ;;^  =  I. 

Formulas  (i)  and  (2)  may  also  be  appHed  to  the  problem  of 
estimating  the  probability  of  the  occurrence  of  an  event  from 
the  concurrent  testimony  of  several  witnesses,  X^,  X^,  .  .  . 
Denote  the  probabilities  that  the  witnesses  tell  the  truth  by 
x^  ^a,  .  .  .  Then,  supposing  them  to  testify  independently, 
the  probability  that  they  will  concur  in  the  truth  concerning 
the  event  is  x^x^  .  .  . ;  while  the  probability  that  they  will  con- 
cur in  the  only  other  alternative,  falsehood,  is  (i  — ;t;i)(i  —x^)  . . . 
The  two  alternatives  are  equally  possible.  Hence  by  equations 
(i)  and  (2) 


p  — ^  "  •  •  • 

'       ^^;r,  ...  +  (l-;irj(l  —  ^,)..  .» 

(I-^,)(I-^,)...  ^3) 

^'-;r,;r,  ...  +  (I  _^J(i_^,)../ 
P^  being  the  probability  for  and  P^  that  against  the  event. 

To  illustrate  (3),  if  the  chances  are  3  to  i  that  X^  tells  the 
truth  and  5  to  i  that  X^  tells  the  truth,  x^  =  3/4,  ;r,  =  5/6,  and 
P^=  15/16;  or,  the  chances  are  15  to  i  that  an  event  occurred 
if  they  agree  in  asserting  that  it  did.* 

*  For  some  interesting  applications  of  equations  (3)  see  note  E  of  Appendix 
to  the  Ninth  Bridgewater  Treatise  by  Charles  Babbage  (London,  1838). 


Art.  8.]  PROBABILITIES    OF    FUTURE    EVENTS.  .  487 

It  is  of  theoretical  interest  to  observe  tliat  if  x^,  x^,  .  .  .  ir 
(3)  are  each  greater  than  1/2,  P^  approaches  unity  as  the 
number  of  witnesses  is  indefinitely  increased. 

Prob.  13.  The  groups  of  numbers  of  one  figure  each,  two  figures 
each,  three  figures  each,  etc.,  which  it  is  possible  to  form  from  the 
nine  digits  i,  2,  ...  9  are  printed  on  cards  and  placed  severally  in 
nine  similar  urns.  What  is  the  probability  that  the  number  777  will 
be  drawn  in  a  single  trial  by  a  person  unaware  of  the  contents  of 
the  urns  ? 

Prob.  14.  How  many  witnesses  whose  credibilities  are  each  3/4 
are  essential  to  make  P^  =  0.999  in  equation  (3)  ? 

Art.  8.    Probabilities  of  Future  Events. 

Equations  (2)  of  Art.  7  may  be  written  in  the  following 
manner:  _ 

Pi        P^  Pm  I 

If /j,  /,,...  /^  are  found  by  observation,  P^,  P^,  .  ,  .  P^  will  ex- 
press the  probabilities  of  the  corresponding  causes  or  their 
effects.  When,  as  in  the  case  of  most  physical  facts,  the  num- 
ber of  causes  and  events  is  indefinitely  great,  the  value  of  any 
/  or  P  in  (i)  becomes  indefinitely  small,  and  the  value  of  2p 
must  be  expressed  by  means  of  a  definite  integral.  Let  x  de- 
note the  probabiHty  of  any  particular  cause,  or  of  the  event  to 
which  it  gives  rise.  Then,  supposing  this  and  all  the  other 
causes  mutually  exclusive,  (i  —  x)  will  be  the  probability 
against  the  event.  Now  suppose  it  has  been  observed  that  in 
(m  -f-  n)  cases  the  event  in  question  has  occurred  m  times  and 
failed  n  times.  The  probability  of  such  a  concurrence  is,  by 
Art.  ^,cx*^{\  —  xYy  where  ^  is  a  constant.  Since  x  is  unknown, 
it  may  be  assumed  to  have  any  value  within  the  limits  o  and  I  \ 
and  all  such  values  are  k  priori  equally  possible.     Put 

y  =  cx'"(l  —■  xy. 
Then  evidently  the  probability  that  x  will  fall  within  any  as- 
signed possible  limits  a  and  b  is  expressed  by  the  fraction 


I  ydx       I  ydx\ 


488  PROBABILITY    AND    THEORY    OF    ERRORS.  [ChAP.  X. 

SO  that  the  probability  of  any  particular  x  is  given  by 


Cx'^ii  —  xydx 


This  may  be  regarded  as  the  antecedent  probability  of  the 
cause  or  event  in  question. 

What  then  is  the  probabiHty  that  in  the  next  {r  -\-  s)  trials 
the  event  will  occur  r  times  and  fail  s  times,  if  no  regard  is  had 
of  the  order  of  occurrence?  If  x  were  known,  the  answer 
would  be  by  Arts.  2  and  5 

^^-e--)-.  (3) 

But  since  x  is  restricted  only  by  the  condition  (2),  the  required 
probability  will  be  found  by  taking  the  product  of  (2)  and  (3) 
and  integrating  throughout  the  range  of  x.  Thus,  calling  the 
required  probabiHty  Q, 

fx'^^Xi  —  xY+'dx 

O'W  •-, ■  (-> 

/  x\\  —  xYdx 
0 

The  definite  integrals  which  appear  here  are  known  as  Gamma 
functions.  They  are  discussed  in  all  of  the  higher  treatises  on 
the  Integral  Calculus.  Applying  the  rules  derived  in  such 
treatises  there  results  * 


(r  ^  s)\{m  ■\-  ry.{n  -\-  sy.{m  -^  n^  I ) ' 
r\s\in\n\{m  -\-  n  -\-  r  -\-  s  -hi)! 


If  regard  is  had  to  the  order  of  occurrence  of  the  event ; 
that  is,  if  the  probability  required  is  that  of  the  event  happen- 
ing r  times  in  succession  and  then  failing  s  times  in  succession, 

*  It  is  a  remarkable  fact  that  formula  (5)  is  true  without  restriction  as  to 
values  of  m,  n,  r,  s.  The  formula  may  be  established  by  elementary  considera- 
tions, as  was  done  by  Prevost  and  Lhuilier,  1795.  See  Todhunter's  History  of 
he  Theory  of   Probability,  pp.  453-457- 


Art.  8.]  probabilities  of  future  events.  48^ 

the  factor  (r  +  ^)!/r!^!  in  (3),  (4),  (5)  must  be  replaced  by 
unity. 

To  illustrate  these  formulas,  suppose  first  that  the  event 
has  happened  m  times  and  failed  no  times.  What  is  the  prob- 
ability that  it  will  occur  at  the  next  trial  ?  In  this  case  (4) 
gives 

Q  =    Cx'^'^'dx  I  fx'^dx  —  {m-\-  \)/{m  +  2). 
0  0 

When  m  is  large  this  probability  is  nearly  unity.  Thus,  the 
sun  has  risen  without  failure  a  great  number  of  times  m ;  the 
probability  that  it  will  rise  to-morrow  is 


(.+i)( 


ml  \     ^    mj  ^    m      m  ^ 


which  is  practically  i. 

Secondly,  suppose  an  urn  contains  white  and  black  balls  in 
an  unknown  ratio.  If  in  ten  trials  7  white  and  3  black  balls 
are  drawn,  what  is  the  probability  that  in  the  next  five  trials 
2  white  and  3  black  balls  will  be  drawn?  The  application  of 
(5)  supposes  the  ratio  of  the  white  and  black  balls  in  the  urn 
to  remain  constant.  This  will  follow  if  the  balls  are  replaced 
after  each  drawing,  or  if  the  number  of  balls  in  the  urn  is  sup- 
posed infinite.     The  data  give 

^  =  7,  ^  =  3»  r  =  2,  J  =  3» 

w  +  ^  =  9>     n-\-s  =  6,     r  +  .y=5,     w-|-«+l  =  ii, 

m-\-n-\-r-\-s~{-i  —  16. 

Thus  by  (5) 

5!9!6!ii! 
^=2!3!7!3!i6!  =  30/9i. 

Suppose  there  are  two  mutually  exclusive  events,  the  first 
of  which  has  happened  m  times  and  the  second  n  times  in 
m  -\-n  trials.  What  is  the  probability  that  the  chance  of  the 
occurrence  of  the  first  exceeds  1/2  ?  The  answer  to  this  ques- 
tion is  given  directly  by  equation  (2)  by  integrating  the  nume- 
rator between  the  specified  limits  of  x.     That  is. 


490  PROBABILITY    AND    THEORY    OF   ERRORS,  [Chap    X. 

Cx'^^i  —  xfdx 

P  =  -. •  (6> 

Cx'^^i  —Xfdx 

0 

Thus,  \i  m=  \  and  n  =o,  P  =  3/4 ;  or  the  odds  are  three  to 
one  that  the  event  is  more  likely  to  happen  than  not.  Simi- 
larly, if  the  event  has  occurred  m  times  in  succession, 

P=  I  -(l/2)'«+\ 

which  approaches  unity  rapidly  with  increase  of  n. 

Art.  9.    Theory  of  Errors. 

The  theory  of  errors  may  be  defined  as  that  branch  of  math- 
ematics which  is  concerned,  first,  with  the  expression  of  the  re- 
sultant effect  of  one  or  more  sources  of  error  to  which  com- 
puted and  observed  quantities  are  subject ;  and,  secondly,  with 
the  determination  of  the  relation  between  the  magnitude  of 
an  error  and  the  probability  of  its  occurrence.  In  the  case  of 
computed  quantities  which  depend  on  numerical  data,  such  as. 
tables  of  logarithms,  trigonometric  functions,  etc.,  it  is  usuall)^ 
possible  to  ascertain  the  actual  values  of-  the  resultant  errors. 
In  the  case  of  observed  quantities,  on  the  other  hand,  it  is  not 
generally  possible  to  evaluate  the  resultant  actual  error,  since 
the  actual  errors  of  observation  are  usually  unknown.  In  either 
case,  however,  it  is  always  possible  to  write  down  a  symbolical, 
expression  which  will  show  how  different  sources  of  error  enter 
and  affect  the  aggregate  error ;  and  the  statement  of  such  ant 
expression  is  of  fundamental  importance  in  the  theory  of  errors. 

To  fix  the  ideas,  suppose  a  quantity  Q  to  be  a  fLinction  of 
several  independent  quantities  x,  y,  z  . .  .\  that  is, 

and  let  it  be  required  to  determine  the  error  in  Q  due  to  errors^ 
in  X,  y,  2  .  .  .  Denote  such  errors  by  z/0,  /Ix,  Ay,  Az  .  .  , 
Then,  supposing  the  errors  so  small  that  their  squares,  prod- 
ucts, and  higher  powers  may  be  neglected,  Taylor's  series  gives 


Art.  10.]  LAWS  OF  error.  491 

This  equation  may  be  said  to  express  the  resultant  actual  error 
of  the  function  in  terms  of  the  component  actual  errors,  since 
the  actual  value  of  AQ  is  known  when  the  actual  errors  of 
Xy  y,  z  .  .  .  are  known.  It  should  be  carefully  noted  that  the 
quantities  x^y^  z  ...  are  supposed  subject  to  errors  which  are 
independent  of  one  another.  The  discovery  of  the  independent 
sources  of  error  is  sometimes  a  matter  of  difficulty,  and  in  general 
requires  close  attention  on  the  part  of  the  student  if  he  would 
avoid  blunders  and  misconceptions.  Every  investigator  in  work 
of  precision  should  have  a  clear  notion  of  the  error-equation  of 
the  type  (i)  appertaining  to  his  work;  for  it  is  thus  only  that 
he  can  distinguish  between  the  important  and  unimportant 
sources  of  error. 

Prob.  15.  Write  out  the  error-equation  in  accordance  with  (i) 
for  the  function  Q  —  xyz  -f  x^  log  {y/z). 

Prob.  16.  In  a  plane  triangle  a/b  =  sin^/sin^.  Find  the  error 
in  a  due  to  errors  in  b,  A^  and  B. 

Prob.  17.   Suppose  in  place  of  the  data  of  problem   16  that  the 
angles  used  in  computation  are  given  by  the  following  equations  : 
^  =  ^,+J(i8o°-^,-  B-  Q,  B  =  B,^  i(i8o°  -  A,-~B^  -  C,), 
where  ^,,  B^y  C,  are  observed  values.     What  then  is  Aa"^ 

Prob.  18.  If  w  denote  the  weight  of  a  body  and  r  the  radius  of 
the  earth,  show  that  for  small  changes  in  altitude,  Aw/w  =  —  Ar/r\ 
whence,  if  a  precision  of  one  part  in  500000000  is  attainable  in  com- 
paring two  nearly  equal  masses,  the  effect  of  a  difference  in  altitude 
of  one  centimeter  in  the  scale-pans  of  a  balance  will  be  noticeable.* 

Art.  10.     Laws  of  Error. 

A  law  of  error  is  a  function  which  expresses  the  relative 
frequency  of  occurrence  of  errors  in  terms  of  their  magnitudes. 
Thus,  using  the   customary  notation,  let  e  denote  the  magni- 

*  This  problem  arose  with  the  International  Bureau  of  Weights  and  Measures^ 
whose  work  of  intercomparison  of  the  Prototype  Kilogrammes  attained  a  pre- 
cision indicated  by  a  probable  error  of  1/500  000  oooth  part  of  a  kilogramme. 


492  PROBABILITY    AND    THEORY    OF    ERRORS.  [ChaP.  X. 

tude  o.  any  error  in  a  system  of  possible  errors.  Then  the  law 
of  such  system  may  be  expressed  by  an  equation  of  the  form 

y  =  0{ey  (I) 

Representing  e  as  abscissa  and  y  as  ordinate,  this  equation 
gives  a  curve  called  the  curve  of  frequency,  the  nature  of  which, 
as  is  evident,  depends  on  the  form  of  the  function  0.  Tliis 
equation  gives  the  relative  frequency  of  occurrence  of  errors  in 
the  system  ;  so  that  if  e  is  continuous  the  probability  of  the 
occurrence  of  any  particular  error  is  expressed  by  yde  =  0(e)^e; 
which  is  infinitesimal,  as  it  plainly  should  be,  since  in  any  con- 
tinuous system  the  number  of  different  values  of  e  is  infinite. 

Consider  the  simplest  form  of  0(e),  namely,  that  in  which 
0(e)  =  c,  a.  constant.  This  form  of  0(e)  obtains  in  the  case  of 
the  errors  of  tabular  logarithms,  natural  trigonometric  func- 
tions, etc.  In  this  case  all  errors  between  minus  a  half-unit 
and  plus  a  half-unit  of  the  last  tabular  place  are  equally  likely 
to  occur.  Suppose,  to  cover  the  class  of  cases  to  which  that 
just  cited  belongs,  all  errors  between  the  limits  —a  and  -\-a 
are  equally  likely  to  occur.  The  probability  of  any  individual 
error  will  then  be  <p(e)de  =  cde,  and  the  sum  of  all  such  prob- 
abilities, by  equation  (2),  Art.  4,  must  be  unity.     That  is, 

/  (p{e)de  =  c  jde  =  I. 

-a  -a 

This  gives  c  =  1/2^,  or  by  (i)  y  =  i/2a.     The  curve  of  fre- 
Q  quency  in  this  case  is  shown  in  the  figure, 

AB  being  the  axis  of  e  and  OQ  that  of  y. 
It  is  evident  from  this  diagram  that  if  the 
errors  of  the  system  be  considered  with 
respect  to  magnitude  only,  half  of  them 
should  be  greater  and  half  less  than  a/2. 
•  This  is  easily  found  to  be  so  in  the  case  of 
tabular  logarithms,  etc. 

As  a  second  illustration  of  (i),  suppose  y  and  e  connected 
by  the  relation  j^'  =  c  Vd"  —  e^  where  a  is  the  radius  of  a  circle, 


Art.  11.]  TYPICAL   ERRORS    OF    A    SYSTEM.  493 

c  a  constant,  and  e  may  have  any  value  between  —  a  and  -f-  a. 
Then  the  condition 


:Jde  V 


c  I  de  ya  —e  =  i 

-  a 

gives  c  =  2/{d7i).  In  this,  as  in  the  preceding  case,  0(-|-  e)  = 
0(—  e),  the  meaning  of  which  is  that  positive  and  negative 
errors  of  the  same  magnitude  are  equally  likely  to  occur.  It 
will  be  noticed,  however,  that  in  the  latter  case  small  errors 
have  a  much  higher  probability  than  those  near  the  limit  a, 
while  in  the  former  case  all  errors  have  the  same  probability. 

In  general,  when  e  is  continuous  0(e)  must  satisfy  the  condi- 
tion   /  ((>{e)de  =  I,  the  limits  being  such  as  to  cover  the  entire 

range  of  values  of  e.  The  cases  most  commonly  met  with  are 
those  in  which  0(e)  is  an  even  function,  or  those  in  which 
0(-|-  e)  =  0(—  e).  In  such  cases,  if  ±  «  denote  the  limiting 
value  of  €, 

fcl>{e)de  =  2f(l>{e)de  =  I.  (3) 

Art.  11.    Typical  Errors  of  a  System. 

Certain  typical  errors  of  a  system  have  received  special 
designations  and  are  of  constant  use  in  the  literature  of  the 
theory  of  errors.  These  special  errors  are  the  probable  error, 
the  mean  error,  and  the  average  error.  The  first  is  that  error 
of  the  system  of  errors  which  is  as  likely  to  be  exceeded  as 
not  ;  the  second  is  the  square  root  of  the  mean  of  the  squares 
of  all  the  errors ;  and  the  third  is  the  mean  of  all  the  errors 
regardless  of  their  signs.  Confining  attention  to  systems  in 
which  positive  and  negative  errors  of  the  same  magnitude 
are  equally  probable,  these  typical  errors  are  defined  mathe- 
matically as  follows.     Let 

Bp  =  the  probable  error, 

e^=  the  mean  error, 

€a  =  the  average  error. 


494  PROBABILITY    AND    THEORY    OF    ERRORS.  [ChAP.  X. 

Then,  observing  (2),  of  Art.  10, 

-e^  0  4-e^  -\-a 

f(p(e)de  =Jct){e)de  =/'0(e)^e  =J(P{e)de  =  |. 

-a  -€^  0  +e^ 

-f-a  +rt 

€„'  =f,p{e)e'de,     6,  =  2j<t>{e)ede. 


(I) 


The  student  should  seek  to  avoid  the  very  common  misap- 
prehension of  the  meaning  of  the  probable  error.  It  is  not 
*'  the  most  probable  error,"  nor  "  the  most  probable  value  of 
the  actual  error";  but  it  is  that  error  which,  disregarding  signs, 
would  occupy  the  middle  place  if  all  the  errors  of  the  system 
were  arranged  in  order  of  magnitude.  A  few  illustrations  will 
suffice  to  fix  the  ideas  as  to  the  typical  errors.  Thus,  take  the 
simple  case  wherein  0(e)  =  ^  =  1/2^,  which  appHes  to  tabular 
logarithms,  etc.     Equations  (i)  give  at  once 

ep=±  -a,     e^=  ±^  V  J,     e^  =  ±  -a. 

For  the  case  of  tabular  values,  a  =  0.5  in  units  of  the  last 
tabular  place.     Hence  for  such  values 

€p=  ±  0.25,     €^=  ±  0.29,     e«  =  ±  0.25. 

Prob.  19.  Find  the  typical  errors  for  the  cases  in  which  the  law 
of  error  is  0(e)  =  cVa'  —  e\  0(e)  =zc{±a^e),  (f>{e)=c  cos'(;re/2«); 
£  being  a  constant  to  be  determined  in  each  case  and  e  having  any 
value  between  —  a  and  +  a. 

Art.  12.    Laws  of  Resultant  Error. 

When  several  independent  sources  of  error  conspire  to  pro- 
duce a  resultant  error,  as  specified  by  equation  (i)  of  Art.  9, 
there  is  presented  the  problem  of  determining  the  law  of  the 
resultant  error  by  means  of  the  laws  of  the  component  errors. 
The  algebraic  statement  of  this  problem  is  obtained  as  follows 
for  the  case  of  continuous  errors  : 

In  the  equation  (i),  Art.  9,  write  for  brevity 


Art.  12.]  LAWS    OF    RESULTANT    ERROR.  495 

and  let  the  laws  of  error  of  e,  e^,  e^,  .  .  .  be  denoted  by  0(e), 
<p^{e^)y  0a(^a)  •  •  .     Then  the  value  of  e  is  given  by  ^  ^^^~ 

e=e,  +  e,+...  (,) 

The  probabilities  of  the  occurrence  of  any  particular  values 
of  e^,  e,,  .  .  .  are  given  by  (p,{e,)de^,  0a(^a)^S,  •  •  •  i  and  the 
probability  of  their  concurrence  is  the  probability  of  the  cor- 
responding value  of  e.  But  since  this  value  may  arise  in  an 
infinite  number  of  ways  through  the  variations  of  e,,  e„  .  .  . 
over  their  ranges,  the  probabiHty  of  e,  or  cf)(e)de,  will  be 
expressed  by  the  integral  of  (pXe^)de^(f)J^e^de^  .  .  .  subject  to 
the  restriction  (i).  This  latter  gives  e,  =  e—  e,  —  e,  .  .  .,  and 
de^  =  de  for  the  multiple  integration  with  respect  to  e„  e„  .  .  . 
Hence  there  results 

^e)de  =  def(p,{e  —  e,  —  e,  —  .  .  .)(p.,{e^)de^ .  .  .  , 


or 


0(e)  =  fct>Ae-e,  -  e,  -  .  .  .)<PA^,)de^fM^,)de,  ...       (2) 

It  is  readily  seen  that  this  formula  will  increase  rapidly  in 
complexity  with  the  number  of  independent  sources  of  error.^ 
For  some  of  the  most  important  practical  applications,  how- 
ever, it  sufifices  to  limit  equation  (2)  to  the  case  of  two  inde- 
pendent sources  of  error,  each  of  constant  probability  within 
assigned  limits.  Thus,  to  consider  this  case,  let  e^  vary  over 
the  range  —  a  to  -\- a,  and  e^  vary  over  the  range  —  ^  to  +  <^. 
Then  by  equation  (2),  Art.  10, 

0,(6,)  =  i/{2a),     0,(ej  =  1/(2^). 
Hence  equation  (2)  becomes 

^(^) = ihp'- 

In  evaluating  this  integral  e,  must  not  surpass  ±  if  and 
€,  =  e  —  e,  must  not  surpass  ±  a.  Assuming  ay  d,  the  limits 
of  the  integral  for  any  value  of  e  =  e^ -\-  e^  lying  between 
—  {a-\-d)  and  —  {a  —  b)  are  —  b  and  +  (e  +  a).     This  fact  is 

*  The  reader  desirous  of  pursuing  this  phase  of  the  sdbject  should  consult 
Bessel's  Untersuchungen  ueber  die  Wahrscheinlichkeit  der  Beobachtungsfehler; 
-Abhandlungen  von  Bessel  (Leipzig,  1876),  Vol.  II. 


496  PROBABILITY    AND    THEORY    OF    ERRORS.  [Chap.  X. 

made  plain  by  a  numerical  example.  For  instance,  suppose 
a  =  S  and  ^  =  3.  Then  —  (a-\-d)  =  —S  and  —  (a—  d)=  —  3. 
Take  e  =  —  6,  a  number  intermediate  to  —  8  and  —  3.  Then 
the  following  are  the  possible  integer  values  of  e^  and  e,  which 
will  produce  e  =  —  6: 

€  €^      e,  limits  of  e, 

-  6  =  -  5  -  I,     -  I  =  +  (e  +  ^), 
=  -  4  -  2, 

=  -3-3,     -3=-^. 
Similarly,  the   limits  of   e,  for  values  of  e  lying  between 
—  {a  ~  b)  and  -\-  {a  —  b)  are  —  b  and  +  ^i   ^"d  the  limits  of 
e,  for  values  of  e  between  -\- {a  —  b)  and  -{-{a-\-b)  are  +  (e  —  a) 
and  -}-  ^.     Hence 


^(^)=i/'^^-5'°''-<'^-*^<^<+^'^ 


b), 


-b 


k  (3) 


«^^)=i/^^-^S*-'  ^''^  +(-*)<^<+(-+^). 


Thus  it  appears  that  in  this  case  the  graph  of  the  resultant 

law  of  error  is  represented  by  the  upper  base  and  the  two  sides 

of   a  trapezoid,  the  lower  base  being  the 

axis  of  e  and  the  Hne  joining  the  middle 

IIIOIII  ,  .  r  ,     N 

pomts  of  the  bases  bemg  the  axis  of  0(e). 
Ill lOIIII 

(See  the  first  fipfure  in  Art.  13.)     The  prop- 
in  I  lOi  nil         ^     .         .,..,,.         ,       ,  .       . 

erties  of  (3),  includmgr  the  determination 
IIIIIIOIIIIII  ^     ,       ,.  'f"  ,        .„ 

of  the  limits,  are  also  illustrated   by  the 

IIIIIIIOIIIIIII    ,.  .,   ^         ,     ^    , 

adiacent  trapezoid  of   numerals  arrans^ed 

IIIIIIIIOIIIIIIII    ■'         ^,         ^    .  ^  , 

to  represent  the  case  wherein  a  =  0.5  and 

^  =  0.3.     The  vertical  scale,  or  that  for  0(6),  does  not,  how- 
ever, conform  exactly  to  that  for  e. 


Art.  13.]  ERRORS  of  interpolated  values.  497 

Prob.  20.  Prove  that  the  values  of  0(e)  as  given  by  equation  (3) 
satisfy  the  condition  specified  in  equation  (3),  Art.  10. 

Prob.  21.  Examine  equations  (3)  for  the  case  wherein  a  =1  b  and 
b  ^=  o;  and  interpret  for  the  latter  case  the  first  and  last  of  (3). 

Prob.  22.  Find  from  (3),  and  (i)  of, Art.  ii,  the  probable  error  of 
the  sum  of  two  tabular  logarithms. 

Art.  13.     Errors  of  Interpolated  Values. 

Case  I. — One  of  the  most  instructive  cases  to  which  formulas 
(3)  of  Art.  12  are  applicable  is  that  of  interpolated  logarithms, 
trigonometric  functions,  etc.,  dependent  on  first  differences. 
Thus,  suppose  that  v^  and  v^  are  two  tabular  logarithms,  and 
that  it  is  required  to  get  a  value  v  lying  t  tenths  of  the  interval 
from  v^  towards  v^.     Evidently 

V  —  v^-{-  {v^  —  v^t=  {i  —  t)v^  +  tv^  ; 
and  hence  if  e,  e^,  e^  denote  the  actual  errors  of  v^  v^,  v^  ,  re- 
spectively, 

^  =  (i-/K  +  /^,.  (I) 

It  is  to  be  carefully  noted  here  that  e  as  given  by  (i)  re- 
quires the  retention  in  v  of  at  least  one  decimal  place  be- 
yond the  last  tabular  place.  For  example,  let  v  =  log  (?4373) 
from  a  5-place  table.  Then  v^  =  4.38686,  7\  =  4.38703^ 
i>"2  —  ^1  =  +0.00017,  /  =  0.3,  and  7/  =4.38691.1.  Likewise,  as 
found  from  a  7-place  table,  e,  =  —0.45,  e.,  =  +  0.37  in  units  of 
the  fifth  place;  and  hence  by  (i)  e=  —0.20.  That  is,  the 
actual  error  oi  v  =  4,38691.1  is  =  0.20,  and  this  is  verified  by 
reference  to  a  7-place  table. 

The  reader  is  also  cautioned  against  mistaking  the  species 
of  interpolated  values  here  considered  for  the  species  common- 
ly used  by  computers,  namely,  that  in  which  the  interpolated 
value  is  rounded  to  the  nearest  unit  of  the  last  tabular  place. 
The  latter  species  is  discussed  under  Case  II  below. 

Confining  attention  now  to  the  class  of  errors  specified  by 
equation  (i),  there  result  in  the  notation  of  the  preceding 
article 

ei  =  (i— /)^i,     e^  =  te^,     and     e  =  e  =  e^-{-e^\ 

and  since  e^  and  e^  each  vary  continuously  between  the  limits 


498 


PROBABILITY  AND  THEORY  OF  ERRORS. 


[Chap.  X. 


±0.5  of  a  unit  of  the  last  tabular  place,  a  and  b  in  equations 
(3)  of  that  article  have  the  values 

a  —  0.5(1  —  t),     b  =0.5^ 

Hence  the  law  of  error  of  the  interpolated  values  is  ex- 
pressed as  follows : 


I 


I  -  t 
0.5  —  e 


for  values  of  e  betw.  —0.5  and  —(0.5—/), 

for  values  of  e  betw.  —(0.5—/)  and  +(0.5—/), 

for  values  of  e  betw.  -\-{p.^—t)  and  +0.5. 


(2) 


-  (I  -  t)t 

The  graph  of  0(e)  for  /  =  1/3  is  shown  by  the  trapezoid 
AB,  BC,  CD  in  the  figure  on  page  500.  Evidently  the  equa- 
tions (2)  are  in  general  represented  by  a  trapezoid,  which  degen- 
erates to  an  isosceles  triangle  when  /  =  1/2. 

The  probable,  mean,  and  average  errors  of  an  interpolated 
value  of  the  kind  in  question  are  readily  found  from  (2),  and 
from  equations  (i)  of  Art.  1 1,  to  be 

e^  =  (i/4)(i  -  /)  for      o  <  /  <  1/3,  '^ 

=  1/2  -  (1/2)1/2/(1  -  /)      for  1/3  <  ^  <  2/3, 

=  1/4/ 


W-(l-2/)-| 
(  96(1  -  t)t  s 
I  -  (I  -  2tY 


lA 


24(1  - 
I  —  {2t 


t)t 


ly 


for  2/z<t  <  1. 


for      o  <t  <  1/2, 


for  1/2  <  t  <  I. 


Y    (3) 


24(1  —  /)/ 

It  is  thus  seen  that  the  probable  error  of  the  interpolated 
value  here  considered  decreases  from  0.25  to  0.15  of  a  unit  of 
the  last  tabular  place  as  t  increases  from  o  to  0.5.  Hence  such 
values  are  more  precise  than  tabular  values  ;  and  the  computer 
who  desires  to  secure  the  highest  attainable  precision  with  a 
given  table  of  logarithms  should  retain  one  additional  figure 
beyond  the  last  tabular  place  in  interpolated  values. 


Art,  13.]        ERRORS  OF  INTERPOLATED  VALUES.  499 

Case  II. — Recurring  to  the  equation  v  =:v^^  t{v^  —  v^  for  an 
interpolated  value  v  in  terms  of  two  consecutive  tabular  values 
z/j  and  z/j,  it  will  be  observed  that  if  the  quantity  t{v^  —  v^  is 
rounded  to  the  nearest  unit  of  the  last  tabular  place,  a  new  error 
is  introduced.  For  example,  if  z/^  =  log  1633  =  3.21299,  and 
^a  =  log  1634  =3.21325  from  a  5-place  table,  v^  —  v^=-\-26 
units  of  the  last  tabular  place  ;  and  if  /  =  1/3,  t{y^  —  v^  =  8f  ^ 
so  that  by  the  method  of  interpolation  in  question  there  results 
V  =  3.21299  +9  =  3.21308.  Now  the  actual  errors  of  v^  and 
v^  are,  as  found  from  a  /-place  table,  —  0.38  and  +  0.21  in  units 
of  the  fifth  place.  Hence  the  actual  error  of  v  is  by  equation 
(0»  f  X  —  0.38  +  i  X  +  0.21  —  J  =  —  0.52,  as  is  shown  di- 
rectly by  a  7-place  table. 

It  appears,  then,  that  in  this  case  the  error-equation  cor- 
responding to  (i)  is 

e=(i-t)e,  +  U,+e,,  (4) 

wherein  e^  and  e^  are  the  same  as  in  (i)  and  ^3  is  the  actual  error 
that  comes  from  rounding  t{v^  —  v^  to  the  nearest  unit  of  the 
last  tabular  place. 

The  error  e^,  however,  differs  radically  in  kind  from  e^  and 
e^.  The  two  latter  are  continuous,  that  is,  they  may  each  have 
any  value,  between  the  limits  —  0.5  and  +0-5  '■>  while  e^  is  dis- 
continuous, being  limited  to  a  finite  number  of  values  depend- 
ent on  the  interpolating  factor  /.  Thus,  for  t  =  1/2  the  only 
possible  values  of  e^  are  o  +  1/2,  and  —  1/2  ;  likewise  for  /  = 
1/3,  the  only  possible  values  of  e^  are  o,  -[-  1/3,  and  —  1/3.  It 
is  also  clear  that  the  maximum  value  of  e,  which  is  constant  and 
equal  to  1/2  for  (i),  is  variable  for  (4)  in  a  manner  dependent 
on  /.     For  example,  in  (4), 

The  maximum  of  e  =  1/2  -[-  1/2  =  i,      for  /  =  1/2, 

«  ^  =  1/2  +  1/3  =  5/6,  "    t  =  1/3, 

«  "  **  ^  =  1/2  +  1/2  =1,       '*    /  =  1/4, 

«  "  ^  =  1/2  +  2/5  =  9/10  "    t  =  1/5. 

The  determination  of  the  law  of  error  for  this  case  presents 
some  novelty,  since  it  is  essential  to  combine  the  continuous 
errors  (i  —  t)e^  and  te^  with  the  discontinuous  error  e^.     The 


500 


PROBABILITY  AND  THEORY  OF  ERRORS. 


[Chap.  X. 


simplest  mode  of  attacking  the  problem  seems  to  be  the  fol- 
lowing quasi-geometrical  one.     In  the  notation  of  Arts.  12  and 
13,  put  in  (4)  ^  =  e,  (I  -  t)e,  =  e„  te,  =  e„  and  e,  =  e,.     Then 
e  =  (e,  +  ej  +  €3.  (S) 

The  law  of  error  for  (e^  +  e,)  is  given  by  equation  (2)  for  any 
value  of  L  Hence  for  a  given  value  of  t  there  will  be  as  many 
expressions  of  0(e)  as  there  are  different  values  of  e,.  The 
graphs  of  0(e)  will  all  be  of  the  same  form  but  will  be  differently 
placed  with  reference  to  the  axis  of  0(e).     Thus,  if  t  =  1/3  the 

values  of  e3  are  —  1/3,  o,  and 
-|-  1/3,  and  these  are  equally 
likely  to  occur.  For  €3  =  0  the 
graph  is  given  directly  by  (2), 
and  is  the  trapezoid  ABCD 
symmetrical  with  respect  to  OQ. 
For  eg  =  —  1/3  the  graph  is 
abQd,  of  the  same  form  as 
ABCD  but  shifted  to  the  left 
by  the  amount  of  £3  =  —  1/3. 
d  D  d'  Similarly,  the  graph  for  the  case 
of  £3  =  +  1/3  is  a'Qh'd\  and  is  produced  by  shifting  ABCD  to 
the  right  by  an  amount  equal  to  +  1/3. 

Now,  since  the  three  systems  of  errors  for  this  case  are 
equally  likely  to  occur,  they  may  be  combined  into  one  system 
by  simple  addition  of  the  corresponding  element  areas  of  the 
several  graphs.  Inspection  of  the  diagram  shows*  that  the 
resultant  law  of  error  is  expressed  by 

0(e)  =  (i/4)(5  +  6e)     for  _  5/6  <  e  <  -  1/6,  ^ 

=  1  for  -  i/6<e<+i/6,  V        (6) 

=  (l/4)(5  -  6e)      for  +  1/6  <  e  <  +  5/6.  J 

This  is  represented  by  a  trapezoid  whose  lower  base  is  10/6, 
upper  base  2/6,  and  altitude  i. 


*  Sum  the  three  areas  and    divide   by  3  to  make   resultant   area  =  i,    as 
required  by  equation  (3),  Art.  10. 


Art.  13.] 


ERRORS    OF    INTERPOLATED    VALUES. 


501 


As  a  second  illustration,  consider  equation  (5)  for  the  case 
/  =  1/2.     In  this  case  e,  must  be  either  o  or  1/2,  the  sign  of 
which  latter  is  arbitrary.     For  €3  =  o,  equations  (2)  give 
0(e)  =  2  +  4e     for     —  1/2  <  e  <  o,  ^ 

=  2  —  4e     for     o  <  e  <-f  1/2.  )  (^^ 

This   function   is   represented  by  the   isosceles  triangle  AQE 

whose  altitude  OQ  is  twice  the  base  AE. 

Similarly  (p{e)  for  c,  =  -|-  1/2  would 

be  represented  by  the  triangle  AQE  dis- 
placed to  the  right  a  distance  1/2  ;  and 
if  the  two  systems  for  ^3  =  0  and  €3  = 
+  1/2  be  combined  into  one  system, 
their  resultant  law  of  error  is  evidently 
0(e)=  i-f-2e     for  —  i/2<e<  o,      \ 

=  I  foro<  e<  +  i/2,      I     (8) 

=  2  —  2e  for  +  1/2  <  e<  I  ;   ) 
the  graph  of  which  is  ABCD.    On  the 

other  hand,  if  the  errors  in  this  combined  system  be  considered 
with  respect  to  magnitude  only,  the  law  of  error  is 

0(e)  =  2(1  -  e)     for     o  <  e<  I,  (9) 

the  graph  of  which  is  OQD. 

The  student  should  observe  that  (6),  (7),  (8),  and  (9)  satisfy 

the   condition    /  (t){e)de  =  I   if  the  integration  embraces   the 
whole  range  of  e. 

The  determination  of  the  general  form  of  (p{e)  in  terms  of 
the  interpolating  factor  /  for  the  present  case  presents  some 
difficulties,  and  there  does  not  appear  to  be  any  published  solu- 
tion of  this  problem.*  The  results  arising  from  one  phase  of 
the  problem  have  been  given,  however,  by  the  author  in  the 
Annals  of  Mathematics,t  and  may  be  here  stated  without 
proof.  The  phase  in  question  is  that  wherein  /  is  of  the  form 
i/n,  n  being  any  positive  integer  less  than  twice  the  greatest 

*  The  author  explained  a  general  method  of  solution  in  a  paper  read  at  the 
summer  meeting  of  the  American  Mathematical  Society,  August,  1895. 
t  Vol.  II,  pp.  54-59. 


502  PROBABILITY    AND    THEORY    OF    ERRORS.  [Chap.  X. 

tabular  difference  of  the  table  to  which  the  formulas  are  ap- 
plied. For  this  restricted  form  of  t  the  possible  maximum 
value  of  e  as  given  by  equation  (5)  is,  in  units  of  the  last 
tabular  place,  {2n  —  \)/n  for  n  odd  and  i  for  ;/  even. 

The  possible  values  of  e^  of  equation  (5)  are 
o, 


I 
I 

2 

n  —  I 

1 

± 

I 
2 

for  n  odd, 
for  n  even. 

•  •  •  ^     211 
n-^2 

•••-^      2n    ' 

o, 

An  important  fact  with  regard  to  the  error  1/2  for  ;/  even 
is  that  its  sign  is  arbitrary,  or  is  not  fixed  by  the  computation 
as  is  the  case  with  all  the  other  errors.  However,  the  com- 
puter's rule,  which  makes  the  rounded  last  figure  of  an  inter- 
polated value  even  when  half  a  unit  is  to  be  disposed  of,  will,, 
in  the  long-run,  make  this  error  as  often  plus  as  minus. 

The  laws  of  error  which  result  are  then  as  follows : 

For  n  odd. 

0(e)  =  1  for  e  between  —  \/2n  and  +  i/2«, 

n     (2n  —  I         \ 
0(e)  = I ±  el  for  ebetw.  ^i/2n  and  =F(2«— 1)/2«. 

For  n  even. 
n  (2n  —  2  \ 

0(e)  =  —. A ±  ^)  for  e  between  o  and  =f  i/«, 

n     (2n  —  I  \ 

-y ±  el  for  e  betw.  hF  i/n  and  =f  («  --  i)/«,. 

—     /  \'(i  ±  e)         for  e  between  ^  (n—  i/n)  and  =p  i. 

2{n  —  ly  ^  \         /    / 

By  means  of  these  formulas  and  (i)  of  Art.  1 1  the  probable, 
mean,  and  average  errors  for  any  value  of  n  can  be  readily 
found.  The  following  table  contains  the  results  of  such  a  com- 
putation for  values  of  n  ranging  from  i  to  10.  The  maximum 
actual  error  for  each  value  of  n  is  also  added.  The  verifica- 
tion of  the  tabular  quantities  will  afford  a  useful  exercise  to  the 
student. 


n  - 
n 


Art.  13  ]  errors  of  interpolated  values.  503 

Typical  Errors  of  Interpolated  Logarithms,  etc. 


Interpolating 
Factor. 
/=  i/« 

Probable  Error. 

Mean  Error. 

Average  Error. 

Maximum 
Actual  Error. 

I 

0.250 

0.289 

0.250 

1/2 

1/2 

.292 

.408 

.333 

1 

1/3 

.256 

.347 

.287 

5/6 

1/4 

.276 

.382 

.313 

I 

1/5 

.268 

.370 

•303 

9/10 

1/6 

=  277 

.385 

•315 

r 

1/7 

.274 

.380 

.311 

13/14 

1/8 

.279 

.3S9 

.318 

I 

1/9 

.278 

.386 

.3.6 

17/18 

i/io 

.281 

.392 

.320 

I 

When  the  interpolating  factor  /  has  the  more  general  fonn 
tn/n,  wherein  m  and  n  are  integers  with  no  common  factor,  the 
possible  values  of  e,  are  the  same  as  for  /  =  i/n.  But  equa- 
tions (3)  of  Art.  12  are  not  the  same  for  t  =  tn/n  as  for^  =  i/n, 
and  hence  for  the  more  general  form  of  /,  0(e)  assumes  a  new 
type  which  is  somewhat  more  complex  than  that  discussed 
above.  The  limits  of  this  work  render  it  impossible  to  extend 
the  investigation  to  these  more  complex  forms  of  0(e).  It  may 
sufifice,  therefore,  to  give  a  .single  instance  of  such  a  function, 
namely,  that  for  which  /  =  2/5.     For  this  case 

0(e)  =1  for  e  between  o  and  ^F  i/io, 

=  (S/^XlS/'fO  ±  ^)  ^<^^  ^  between  =F  i/io  and  q:  3/10, 
=  (5/3X4/5  ±  e)       for  e  between  q=  3/10  and  =F  7/10, 
=  (5/6)(9/iO  ±  e)     for  e  between  hF  7/io  and  :f  9/10. 
The  graph  of  the  right-hand  half  of  ^     b 
this  function  is  shown  in  the  accompany- 
ing  diagram,    the   whole   graph    being 
symmetrical  with  respect  to  OA^  or  the 
axis  of  0(e). 

Attention  may  be  called  to  the  strik- 
ing resemblance  of  this  graph  to  that  of 
the  law  of  error  of  least  squares. 

Prob.  23.  Show  from  equations  (3)  that  e^  varies  from  1//12 
=  0.29  — ,  for  /  =  o,  to  i/'^^24  =  0.20  -{-,  for  /  =  0.5  ;  and  that  e^ 
varies  from  0.25  to  1/6  for  the  same  limits. 


504  PROBABILITY    AND    THEORY    OF   ERRORS.  [ChAP.  X. 

Prob.  24.  Show  that  the  probable,  mean,  and  average  errors 
for  the  case  of  /  =  2/5  cited  above  (p.  503)  are  ±  0.261,  ±  0.251, 
and  ±  0.290,  respectively. 

Art.  14.  Statistical  Test  of  Theory. 
A  statistical  test  of  the  theory  developed  in  Art.  13  may 
be  readily  drawn  from  any  considerable  number  of  actual  er- 
rors of  interpolated  values  dependent  on  the  same  interpolating 
factor.  The  application  of  such  a  test,  if  carried  out  fully  by 
the  student,  will  go  far  also  towards  fixing  clear  notions  as  to 
the  meaning  of  the  critical  errors. 

Consider  first  the  case  in  which  an  interpolated  value  falls 
midway  between  two  consecutive  values,  and'  suppose  this 
interpolated  value  retains  two  additional  figures  beyond  the 
last  tabular  place.  Then  by  equations  (2),  Art.  13,  the  law  Oif 
error  of  this  interpolated  value  is 

0(e)  =  2  -[-  4e  for  e  between  —  0.5  and  O 
=  2  —  4e  for  e  between  o  and  -|-  0.5. 

Hence  by  equation  (i)  of  Art.  1 1,  or  equation  (3)  of  Art.  12,  the 
probable  error  in  this  system  of  errors  is  i  —  {\)  V2  =  o.i^. 
It  follows,  therefore,  that  in  any  large  number  of  actual  errors 
of  this  system,  half  should  be  less  and  half  greater  than  0.15. 
Similarly,  of  the  whole  number  of  such  errors  the  percentage 
falling  between  the  values  0.0  and  0.2  should  be 

+o.a  +o.a 

J  0(e) ^e  =  2J  {2  —  4e)de  =  0.64 ; 

-0.2  0 

that  is,  sixty-four  per  cent  of  the  errors  in  question  should  be 
less  numerically  than  o.  2. 

To  afford  a  more  detailed  comparison  in  this  case,  the  act- 
ual errors  of  five  hundred  interpolated  values  from  a  5-place 
table  have  been  computed  by  means  of  a  /-place  table.  The 
arguments  used  were  the  following  numbers  :  20005,  20035, 
20065,  20105,  20135,  etc.,  in  the  same  order  to  36635.  The 
actual  and  theoretical  percentages  of  the  whole  number  of 
errors  falling  between  the  limits  0.0  and  o.i,  o.i  and  0.2,  etc., 
are  shown  in  the  tabular  form  following : 


Art.  14.]  statistical  test  of  theory.  505 

T  .     .        r  t:-  Actual  Theoretical 

Limits  of  Errors.  Percentage.  Percentage. 

o.oando.i 33.2  36 

0.1  and  0.2 30.2  28 

0.2  and  0.3 19.0  20 

0.3  and  0.4 13.2  12 

0.4  and  o.  5 4.4  4 

0.0  and  0.1 5 51.4  50 

The  agreement  shown  here  between  the  actual  and  theoretical 
percentages  is  quite  close,  the  maximum  discrepancy  being  2.8 
and  the  average  1.5  per  cent. 

Secondly,  consider  the  case  of  interpolated  mid-values  of  the 
species  treated  under  Case  II  of  Art.  13.  The  law  of  error  for 
this  case  is  given  by  the  single  equation  (9)  of  Art.  13,  namely, 
0(e)  =  2(1  —  e),  no  regard  being  paid  to  the  signs  of  the  errors. 
The  probable  error  is  then  found  from 


whence  6^=  i  —  i  V2  =  0.29.  Similarly,  the  percentage  of 
the  whole  number  of  errors  which  may  be  expected  to  lie,  for 
example,  between  0.0  and  0.2  in  this  system  is 


o.a 


2   /  (l  —  €)d€  =  0.36. 


Using  the  same  five  hundred  interpolated  values  cited 
above,  but  rounding  them  to  the  nearest  unit  of  the  last  tabu- 
lar place  and  computing  their  actual  errors  by  means  of  a  /-place 
table,  the  following  comparison  is  afforded  : 

,  .     .        ,  T-  Actual  Theoretical 

Limits  of  Errors.  Percentage.  Percentage. 

0.0  and  0.2 35.8  36 

0.2  and  0.4 27.8  28 

0.4  and  0.6 18.6  20 

0.6  and  0.8 12.2  12 

0.8  and  i.o 5-6  4 

0.0  and  0.29 49-^  5^ 


506  PROBABILITY    AND    THEORY    OF    ERRORS.  [Chap.  X. 

The  agreement  shown  here  between  the  actual  and  theoretical 
percentages  is  somewhat  closer  than  in  the  preceding  case,  the 
maximum  discrepancy  being  only  1.6  and  the  average  only  0.6 
per  cent. 

Finally,  the  following  data  derived  from  one  thousand  act- 
ual errors  may  be  cited.  The  errors  of  one  hundred  inter- 
polated values  rounded  to  the  nearest  unit  of  the  last  tabular 
place  were  computed  *  for  each  of  the  interpolating  factors 
O.I,  0.2,  .  .  .  0.9.  The  averages  of  these  several  groups  of  act- 
ual errors  are  given  along  with  the  corresponding  theoretical 
errors  in  the  parallel  columns  below: 

Interpolating  Actual  Theoretical 

Factor.  Average  Error.      Average  Error. 

0.1 0.338  0.320 

0.2 0.288  0.303 

0.3 0.321  0.304 

0.4 0.268  0.290 

0.5 0.324  0.333 

0.6 ...    0.276  0.290 

0.7 0.321  0.304 

0.8 0.289  0.303 

0.9 0.347  0.320 

The  average  discrepancy  between  the  actual  and  theoret- 
ical values  shown  here  is  0.017.  It  is,  perhaps,  somewhat 
smaller  than  should  be  expected,  since  the  computation  of  the 
actual  errors  to  three  places  of  decimals  is  hardly  warranted 
by  the  assumption  of  dependence   on    first    differences  only. 

The  average  of  the  whole  number  of  actual  errors  in  this 
case  is  0.308,  which  agrees  to  the  same  number  of  decimals 
with  the  average  of  the  theoretical  errors,  f 

*  By  Prof.  H.  A.  Howe.  See  Annals  of  Mathematics,  Vol.  Ill,  p.  74. 
The  theoretical  averages  were  furnished  to  Prof.  Howe  by  the  author. 

f  The  reader  who  is  acquainted  with  the  elements  of  the  method  of  least 
squares  will  find  it  instructive  to  apply  that  method  to  equation  (i),  Art.  13, 
and  derive  the  probable  error  of  e.     This  is  frequently  done  without  reserve  by 


Art.  14.]  statistical  test  of  theory.  507 

Prob.  25.  Apply  formulas  (3)  of  Art.  12  to  the  case  of  the  sum 
or  difference  of  two  tabular  logarithms  and  derive  the  correspond- 
ing values  of  the  probable,  mean,  and  average  errors.  The  graph 
of  0(e)  is  in  this  case  an  isosceles  triangle  whose  base,  or  axis  of  e, 
is  2,  and  whose  altitude,  or  axis  of  0(e),  is  i. 

those  familiar  with  least  squares.     Thus,  the  probable  error  of  ei  or  <?2  being 
0.25,  the  probable  error  of  e  is  found  to  be 


0.25  r  I  —  2/  +  2iK 

This  varies  between  0.25  for  ^  =  o  and  0.18  for  /  =  |^ ;  while  the  true  value  of 
the  probable  error,  as  shown  by  equations  (3),  Art.  13,  varies  from  0,25  to  0.15 
for  the  same  values  of  t.  It  is,  indeed,  remarkable  that  the  method  of  least 
squares,  which  admits  infinite  values  for  the  actual  errors  ei  and  e^,  should  give 
so  close  an  approximate  formula  as  the  above  for  the  probable  error  of  ^. 

Similarly,  one  accustomed  to  the  method  of  least  squares  would  be  inclined 
to  apply  it  to  equation  (4).  Art.  13,  to  determine  the  probable  error  of  e.  The 
natural  blunder  in  this  case  is  to  consider  ei,  ^7 ,  and  ea  independent,  and  ea  like 
ei  and  e<i  continuous  betweer  the  limits  0.0  and  0.5  ;  and  to  assign  a  probable 
error  of  0.25  to  each.     In  t'.is  manner  the  value 


0.25^2(1-/4-/2) 

is  derived.  But  this  is  absurd,  since  it  gives  0.25  V2  instead  of  0.25  for  /  =  o. 
The  formula  fails  then  to  give  even  approximate  results  except  for  values  of  / 
near  0.5. 


-5U8  HISTORY    OF    MODERN    MATHEMATICS.  [ChaP.  XI 


Chapter  XI. 

HISTORY  OF  MODERN  MATHEMATICS. 

By  David  Eugene  Smith, 
Professor  of  Mathematics  in  the  Michigan  State  Normal  School. 

Art.  1.    Introduction. 

Modern  Mathematics  is  a  term  by  no  means  well  defined. 
Algebra  cannot  be  called  modern,  and  yet  the  theory  of  equa- 
tions has  received  some  of  its  most  important  additions  during 
the  nineteenth  century,  while  the  theory  of  forms  is  a  recent 
creation.  Similarly  with  elementary  geometry;  the  labors  of 
Lobachevsky  and  Bolyai  during  the  second  quarter  of  the 
century  threw  a  new  light  upon  the  whole  subject,  and  piore 
recently  the  study  of  the  triangle  has  added  another  chapter 
to  the  theory.  Thus  the  history  of  modern  mathematics  must 
-also  be  the  modern  history  of  ancient  branches,  while  subjects 
which  seem  the  product  of  late  generations  have  root  in  other 
•centuries  than  the  present. 

How  unsatisfactory  must  be  so  brief  a  sketch  may  be  in- 
ferred from  a  glance  at  the  Index  du  Repertoire  Bibliographique 
des  Sciences  Math^matiques  (Paris,  1893),  whose  seventy-one 
pages  contain  the  mere  enumeration  of  subjects  in  large  part 
modern,  or  from  a  consideration  of  the  twenty-six  volumes  of  the 
Jahrbuch  iiber  die  Fortschritte  der  Mathematik,  which  now 
devotes  over  a  thousand  pages  a  year  to  a  record  of  the  pro- 
gress of  the  science.^ 

The  seventeenth  and  eighteenth  centuries  laid  the  founda- 

*  The  foot-notes  give  only  a  few  of  the  authorities  which  might  easily  be 
cited.  They  are  thought  to  include  those  from  which  considerable  extracts 
have  been  made,  the  necessary  condensation  of  these  extracts  making  any  other 
iorm  of  acknowledgment  impossible. 


Art.  1.]  INTRODUCTION.  509' 

tions  of  much  of  the  subject  as  known  to-day.  The  discovery" 
of  the  analytic  geometry  by  Descartes,  the  contributions  to  the 
theory  of  numbers  by  Fermat,  to  algebra  by  Harriot,  to 
geometry  and  to  mathematical  physics  by  Pascal,  and  the 
discovery  of  the  differential  calculus  by  Newton  and  Leibniz, 
all  contributed  to  make  the  seventeenth  century  memorable. 
The  eighteenth  century  was  naturally  one  of  great  activity. 
Euler  and  the  Bernoulli  family  in  Switzerland,  d'Alembert, 
Lagrange,  and  Laplace  in  Paris,  and  Lambert  in  Germany,, 
popularized  Newton's  great  discovery,  and  extended  both  its 
theory  and  its  applications.  Accompanying  this  activity,  how- 
ever, was  a  too  implicit  faith  in  the  calculus  and  in  the  in- 
herited principles  of  mathematics,  which  left  the  foundations 
insecure  and  necessitated  their  strengthening  by  the  succeed- 
ing generation. 

The  nineteenth  century  has  been  a  period  of  intense  study 
of  first  principles,  of  the  recognition  of  necessary  limitations 
of  various  branches,  of  a  great  spread  of  mathematical  knowl- 
edge, and  of  the  opening  of  extensive  fields  for  applied  mathe- 
matics. Especially  influential  has  been  the  establishment  of 
scientific  schools  and  journals  and  university  chairs.  The 
great  renaissance  of  geometry  is  not  a  little  due  to  the  founda- 
tion of  the  Ecole  Polytechnique  in  Paris  (1794-5),  and  the  simi- 
lar schools  in  Prague  (1806),  Vienna  (181 5),  Berlin  (1820), 
Karlsruhe  (1825),  and  numerous  other  cities.  About  the  mid- 
dle of  the  century  these  schools  began  to  exert  a  still  a  greater 
influence  through  the  custom  of  calling  to  them  mathemati- 
cians of  high  repute,  thus  making  Zurich,  Karlsruhe,  Munich, 
Dresden,  and  other  cities  well  known  as  mathematical  centers. 

In  1796  appeared  the  first  number  of  the  Journal  de  TEcole 
Polytechnique.  Crelle's  Journal  fiir  die  reine  und  angewandte 
Mathematik  appeared  in  1826,  and  ten  years  later  Liouville 
began  the  publication  of  the  Journal  de  Mathematiques  pures 
et  appliquees,  which  has  been  continued  by  Resal  and  Jordan. 
The  Cambridge  Mathematical  Journal  was  established  in  1839, 
and   merged   into   the  Cambridge  and    Dublin    Mathematical 


510  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XI. 

Journal  in  1846.  Of  the  other  periodicals  which  have  contrib- 
uted to  the  spread  of  mathematical  knowledge,  only  a  few 
can  be  mentioned  :  the  Nouvelles  Annales  de  Mathematiques 
(1842),  Grunert's  Archiv  der  Mathematik  (1843),  Tortolini's 
Annali  di  Scienze  Matematiche  e  Fisiche  (1850),  Schlomilch's 
Zeitschrift  fur  Mathematik  und  Physik  (1856),  the  Quarterly 
Journal  of  Mathematics  (1857),  Battaglini's  Giornale  di  Mate- 
matiche (1863),  the  Mathematische  Annalen  (1869),  the  Bulle- 
tin des  Sciences  Mathematiques  (1870),  the  American  Jour- 
nal of  Mathematics  (1878),  the  Acta  Mathematica  (1882),  and 
the  Annals  of  Mathematics  (1884).*  To  this  list  should  be 
added  a  recent  venture,  unique  in  its  aims,  namely,  L'lnter- 
m^diaire  des  Mathematiciens  (1894),  and  two  annual  publica- 
tions of  great  value,  the  Jahrbuch  already  mentioned  (1868), 
and  the  Jahresbericht  der  deutschen  Mathematiker-Vereini- 
gung(i892). 

To  the  influence  of  the  schools  and  the  journals  must  be 
added  that  of  the  various  learned  societies  f  whose  published 
proceedings  are  widely  known,  together  with  the  increasing 
liberality  of  such  societies  in  the  preparation  of  complete 
works  of  a  monumental  character. 

The  study  of  first  principles,  already  mentioned,  was  a  nat- ' 
ural  consequence  of  the  reckless  application  of  the  new  cal- 
culus  and    the    Cartesian    geometry    during    the    eighteenth 
century.     This  development  is  seen  in  theorems  relating  to  in- 
finite series,  in  the  fundamental  principles  of  number,  rational, 

*  For  a  list  of  current  mathematical  journals  see  the  Jahrbuch  iiber  die  Fort- 
schritte  der  Mathematik.  A  small  but  convenient  list  of  standard  periodicals  is 
given  in  Cart's  Synopsis  of  Pure  Mathematics,  p.  843  ;  Macka}'.  J.  S.,  Notice 
sur  le  journalisme  math^matique  en  Angleterre,  Association  frangaise  pour 
I'Avancement  des  Sciences,  1893,  II,  303  ;  CajoH,  F.,  Teaching  and  History  of 
Mathematics  in  the  United  States,  pp.  94,  277 ;  Hart,  D.  S.,  History  of  Ameri- 
can Mathematical  Periodicals,  The  Analyst,  Vol.  II,  p.  131. 

f  For  a  list  of  such  societies  consult  any  recent  number  of  the  Philosophical 
Transactions  of  Royal  Society  of  London.  Dyck,  W.,  Einleitung  zu  dem  flir 
den  mathematischen  Tell  der  deutschen  Universitatsausstellung  ausgegebenen 
Specialkatalog,  Mathematical  Papers  Chicago  Congress  (Nevjr  York,  1896),  p.  44. 


Art.  2.]  THEORY  of  numbers.  5]  1 

irrational,  and  complex,  and  in  the  concepts  of  limit,  conti- 
unity,  function,  the  infinite,  and  the  infinitesimal.  But  the 
nineteenth  century  has  done  more  than  this.  It  has  created 
new  and  extensive  branches  of  an  importance  which  promises 
much  for  pure  and  applied  mathematics.  Foremost  among 
these  branches  stands  the  theory  of  functions  founded  by 
Cauchy,  Riemann,  and  Weierstrass,  followed  by  the  descrip- 
tive and  projective  geometries,  and  the  theories  of  groups,  of 
forms,  and  of  determinants. 

The  nineteenth  century  has  naturally  been  one  of  speciaHz- 
ation.  At  its  opening  one  might  have  hoped  to  fairly  compass 
the  mathematical,  physical,  and  astronomical  sciences,  as  did 
Lagrange,  Laplace,  and  Gauss.  But  the  advent  of  the  new 
generation,  with  Monge  and  Carnot,  Poncelet  and  Steiner, 
Galois,  Abel,  and  Jacobi,  tended  to  split  mathematics  into 
branches  between  which  the  relations  were  long  to  remain  ob- 
scure. In  this  respect  recent  years  have  seen  a  reaction,  the 
unifying  tendency  again  becoming  prominent  through  the 
theories  of  functions  and  groups.* 

Art.  2.    Theory  of  Numbers. 

The  Theory  of  Numbers,f  a  favorite  study  among  the 
Greeks,  had  its  renaissance  in  the  sixteenth  and  seventeenth 
centuries  in  the  labors  of  Viete,  Bachet  de  Meziriac,  and  es- 
pecially Fermat.  In  the  eighteenth  century  Euler  and 
Lagrange  contributed  to  the  theory,  and  at  its  close  the  sub- 
ject began  to  take  scientific  form  through  the  great  labors  of 
Legendre  (1798),  and  Gauss  (1801).  With  the  latter's  Disquisi- 
tiones  Arithmeticae  (1801)  may  be  said  to  begin  the  modern 
theory  of  numbers.  This  theory  separates  into  two  branches, 
the  one  deahng  with  integers,  and  concerning  itself  especially 

*  Klein,  F.,  The  Present  State  of  Mathematics,  Mathematical  Papers  of 
Chicago  Congress  (New  York,  1896),  p.  133. 

f  Cantor,  M.,  Geschichte  der  Mathematik,  Vol.  Ill,  p.  94;  Smith,  H.  J.  S., 
Report  on  the  theory  of  numbers;  Collected  Papers,  Vol.  I;  Stolz,  O.,  Gros- 
sen  und  Zahlen,  Leipzig,  1891. 


512  HISTORY    OF    MODERN    MATHEMATICS.  [ChaP.  XI. 

with  (i)  the  study  of  primes,  of  congruences,  and  of  residues, 
and  in  particular  with  the  law  of  reciprocity,  and  (2)  the  theory 
of  forms,  and  the  other  dealing  with  complex  numbers. 

The  Theory  of  Primes*  has  attracted  many  investigators 
during  the  nineteenth  century,  but  the  results  have  been  de- 
tailed rather  than  general.  Tchebichef  (1850)  was  the  first  to 
reach  any  valuable  conclusions  in  the  way  of  ascertaining  the 
number  of  primes  between  two  given  limits.  Riemann  (1859) 
also  gave  a  well-known  formula  for  the  limit  of  the  number  of 
primes  not  exceeding  a  given  number. 

The  Theory  of  Congruences  may  be  said  to  start  with 
Gauss's  Disquisitiones.  He  introduced  the  symbolism  a^b 
(mod  c),  and  explored  most  of  the  field.  Tchebichef  published 
in  1847  ^  work  in  Russian  upon  the  subject,  and  in  France 
Serret  has  done  much  to  make  the  theory  known. 

Besides  summarizing  the  labors  of  his  predecessors  in  the 
theory  of  numbers,  and  adding  many  original  and  noteworthy 
contributions,  to  Legendre  may  be  assigned  the  fundamental 
theorem  which  bears  his  name,  the  Law  of  Reciprocity  of  Quad- 
ratic Residues.  This  law,  discovered  by  induction  by  Euler, 
was  enunciated  by  Legendre  and  first  proved  in  his  Theorie 
des  Nombres  (1798)  for  special  cases.  Independently  of  Euler 
and  Legendre,  Gauss  discovered  the  law  about  1795,  and  was 
the  first  to  give  a  general  proof.  To  the  subject  have  also 
contributed  Cauchy,  perhaps  the  most  versatile  of  French 
mathematicians  of  the  century;  Dirichlet,  whose  Vorlesungen 
liber  Zahlentheorie,  edited  by  Dedekind,  is  a  classic  ;  Jacobi, 
who  introduced  the  generalized  symbol  which  bears  his  name  ; 
Liouville,  Zeller,  Eisenstein,  Kummer,  and  Kronecker.  The 
theory  has  been  extended  to  include  cubic  and  biquadratic 
reciprocity,  notably  by  Gauss,  by  Jacobi,  who  first  proved  the 
law  of  cubic  reciprocity,  and  by  Kummer. 

*  Brocard,  H.,  Sur  la  frequence  et  la  totality  des  nombres  premiers;  Nou- 
velle  Correspondence  de  Mathematiques,  Vols.  Vand  VI;  gives  ecent  history  to 
1979. 


Art.  3.]  IRRATIONAL    AND    TRANSCENDENT    NUMBERS.  513 

To  Gauss  is  also  due  the  representation  of  numbers  by 
binary  quadratic  forms.  Cauchy,  Poinsot  (1845),  Lebesgues 
(1859,  1868),  and  notably  Hermite  have  added  to  the  subject. 
In  the  theory  of  ternary  forms  Eisenstein  has  been  a  leader, 
and  to  him  and  H.  J.  S.  Smith  is  also  due  a  noteworthy  ad- 
vance in  the  theory  of  fornis  in  general.  Smith  gave  a  com- 
plete classification  of  ternary  quadratic  forms,  and  extended 
Gauss's  researches  concerning  real  quadratic  forms  to  complex 
forms.  The  investigations  concerning  the  representation  of 
numbers  by  the  sum  of  4,  5,  6,  7,  8  squares  were  advanced  by 
Eisenstein  and  the  theory  was  completed  by  Smith. 

In  Germany,  Dirichlet  was  one  of  the  most  zealous  workers 
in  the  theory  of  numbers,  and  was  the  first  to  lecture  upon  the 
subject  in  a  German  university.  Among  his  contributions  is 
the  extension  of  Fermat's  theorem  on  x^  +j^"  =  -S"",  which  Euler 
and  Legendre  had  proved  for  ;z  =  3,  4,  Dirichlet  showing,  that 
x^ -\- y" ■:^  az^ .  Among  the  later  French  writers  are  Borel ; 
Poincar^,  whose  memoirs  are  numerous  and  valuable  ;  Tannery, 
and  Stieltjes.  Among  the  leading  contributors  in  Germany 
are  Kronecker,  Kummer,  Schering,  Bachmann,  and  Dedekind. 
In  Austria  Stolz's  Vorlesungen  iiber  allgemeine  Arithmetik 
(1885-86),  and  in  England  Mathews'  Theory  of  Numbers 
(Part  I,  1892)  are  among  the  most  scholarly  of  general  works. 
Genocchi,  Sylvester,  and  J.  W.  L.  Glaisher  have  also  added  to 
the  theory. 

Art.  3.    Irrational  and  Transcendent  Numbers. 

The  sixteenth  century  saw  the  final  acceptance  of  negative 
numbers,  integral  and  fractional.  The  seventeenth  century 
saw  decimal  fractions  with  the  modern  notation  quite  generally 
used  by  mathematicians.  The  next  hundred  years  saw  the 
imaginary  become  a  powerful  tool  in  the  hands  of  De  Moivre, 
and  especially  of  Euler.  For  the  nineteenth  century  it  re- 
mained to  complete  the  theory  of  complex  numbers,  to  separate 
irrationals  into  algebraic  and  transcendent,  to  prove  the  exist- 
ence of  transcendent  numbers,  and  to  make  a  scientific  study 


514  HISTORY    OF    MODERN    MATHEMATICS.  [ChAP.  XI. 

of  a  subject  which  had  remained  almost  dormant  since  EucHd, 
the  theory  of  irrationals.  The  year  1872  saw  the  publication 
of  the  theories  of  Weierstrass  (by  his  pupil  Kossak),  Heine 
(Crelle,  74),  G.  Cantor  (Annalen,  5),  and  Dedekind.  Meray  had 
taken  in  1869  the  same  point  of  departure  as  Heine,  but  the 
theory  is  generally  referred  to  the  year  1872.  Weierstrass's 
method  has  been  completely  set  forth  by  Pincherle  (1880),  and 
Dedekind's  has  received  additional  prominence  through  the 
author's  later  work  (1888)  and  the  recent  indorsement  by  Tan- 
nery (1894).  Weierstrass,  Cantor,  and  Heine  base  their  the- 
ories on  infinite  series,  while  Dedekind  founds  his  on  the  idea 
of  a  cut  (Schnitt)  in  the  system  of  real  numbers,  separating  all 
rational  numbers  into  two  groups  having  certain  characteristic 
properties.  The  subject  has  received  later  contributions  at  the 
hands  of  Weierstrass,  Kronecker  (Crelle,  loi),  and  M^ray. 

Continued  Fractions,  closely  related  to  irrational  numbers 
(and  due  to  Cataldi,  1613),"^  received  attention  at  the  hands  of 
Euler,  and  at  the  opening  of  the  nineteenth  century  were 
brought  into  prominence  through  the  writings  of  Lagrange, 
Other  noteworthy  contributions  have  been  made  by  Drucken- 
muUer  (1837),  Kunze  (1857),  Lemke  (1870),  and  Gunther  (1872). 
Ramus  (1855)  first  connected  the  subject  with  determinants, 
resulting,  with  the  subsequent  contributions  of  Heine,  Mobius, 
and  Gunther,  in  the  theory  of  Kettenbruchdeterminanten. 
Dirichlet  also  added  to  the  general  theory,  as  have  numerous 
contributors  to  the  applications  of  the  subject. 

Transcendent  Numbers  f  were  first  distinguished  from  alge- 
braic irrationals  by  Kronecker.  Lambert  proved  (1761)  that 
n  cannot  be  rational,  and  that  e^  {n  being  zero  or  rational)  is 
irrational,  a  proof,  however,  which  left  much   to  be  desired. 

*But  see  Favaro,  A.,  Noti^ie  storiche  sulle  frazioni  continue  dal  secolo  deci- 
moterzo  al  decimosettimo,  Boncompagni's  Bulletino,  Vol.  VII,  1874,  pp.  451, 
533. 

f  Klein,  F.,  VortrSge  ubtr  ausgewShlte  Fragen  der  Elementargeometrie, 
1895,  p.  38  ;  Bachraan,  P.,  Vorlesungen  iiber  die  Natur  der  Irrationalzahlen, 
i8q2. 


Art.  4.]  COMPLEX  numbers.  515 

Legendre  (1794)  completed  Lambert's  proof,  and  showed  that 
Tt  is  not  the  square  root  of  a  rational  number.  Liouville  (1840) 
showed  that  neither  e  nor  e^  can  be  a  root  of  an  integral  quadratic 
equation.  But  the  existence  of  transcendent  numbers  was  first 
established  by  Liouville  (1844,  185 1),  the  proof  being  subse- 
quently displaced  by  G.  Cantor's  (1873).  Hermite  (1873)  first 
proved  e  transcendent,  and  Lindemann  (1882),  starting  from 
Hermite's  conclusions,  showed  the  same  for  7t,  Lindemann's 
proof  was  much  simpHfied  by  Weierstrass  (1885),  still  further 
by  Hilbert  (1893),  and  has  finally  been  made  elementary  by 
Hurwitz  and  Gordan. 

Art.  4.    Complex  Numbers. 

The  Theory  of  Complex  Numbers*  may  be  said  to  have 
attracted  attention  as  early  as  the  sixteenth  century  in  the 
recognition,  by  the  Italian  algebraists,  of  imaginary  or  impos- 
sible roots.  In  the  seventeenth  century  Descartes  distin- 
guished between  real  and  imaginary  roots,  and  the  eighteenth 
saw  the  labors  of  De  Moivre  and  Euler.  To  De  Moivre  is 
due  (1730)  the  well-known  formula  which  bears  h  is  name, 
(cos  (p  -{-t  sin  (py  =  cos  ncp  +  i  sin  mp,  and  to  Euler  (1748)  the 
formula  cos  0  +  ^  sin  0  =  e^'. 

The  geometric  notion  of  complex  quantity  now  arose,  and 
as  a  result  the  theory  of  complex  numbers  received  a  notable 
expansion.  The  idea  of  the  graphic  representation  of  cogiplex 
numbers  had  appeared,  however,  as  early  as  1685,  in  Wallis's 
De  Algebra  tractatus.  In  the  eighteenth  century  Kiihn  (1750) 
and  Wessel  (about  1795)  made  decided  advances  towards  the 
present  theory.  Wessel's  memoir  appeared  in  the  Proceed- 
ings of  the  Copenhagen  Academy  for  1799,  and  is  exceedingly 

*Riecke,  F.,  Die  Rechnung  mit  Richtungszahlen,  1856,  p.  161  ;  Hankel,  H., 
Theorie  derkomplexen  Zahlensysteme,  Leipzig,  1867  ;  Holzmiiller,  G.,  Theorie 
derisogonalen  Verwandtschaften,  1882,  p.  2i;  Macfarlane,  A,,  The  Imaginary 
of  Algebra,  Proceedings  of  American  Association  1892,  p.  33  ;  Baitzer,  R., 
Einfiihrung  der  komplexen  Zahlen,  Crelle,  1882  ;  Stolz,  O.,  Vorlesungen  iiber 
allgemeine  Arithmetik,  2.  Theil,  Leipzig,  1886. 


516  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XL 

clear  and  complete,  even  in  comparison  with  modern  works. 
He  also  considers  the  sphere,  and  gives  a  quaternion  theory 
from  which  he  develops  a  complete  spherical  trigonometry. 
In  1804  the  Abbe  Buee  independently  came  upon  the  same 
idea  which  Wallis  had  suggested,  that  ±  |/_  j  should  repre- 
sent a  unit  line,  and  its  negative,  perpendicular  to  the  real  axis. 
Buee's  paper  was  not  published  until  1806,  in  which  year  Ar- 
gand  also  issued  a  pamphlet  on  the  same  subject.  It  is  to 
Argand's  essay  that  the  scientific  foundation  for  the  graphic 
representation  of  complex  numbers  is  now  generally  referred. 
Nevertheless,  in  1831  Gauss  found  the  theory  quite  unknown, 
and  in  1832  published  his  chief  memoir  on  the  subject,  thus 
bringing  it  prominently  before  the  mathematical  world.  Men- 
tion should  also  be  made  of  an  excellent  little  treatise  by 
Mourey  (1828),  in  which  the  foundations  for  the  theory  of  di- 
rectional numbers  are  scientifically  laid.  The  general  accept- 
ance of  the  theory  is  not  a  little  due  to  the  labors  of  Cauchy 
and  Abel,  and  especially  the  latter,  who  was  the  first  to  boldly 
use  complex  numbers  with  a  success  that  is  well  known. 

The  common  terms  used  in  the  theory  are  chiefly  due  ta 
the  founders.  Argand  called  cos  (p-\-t  sin  0  the  **  direction 
factor",  and  r  =  Va'  +  d'  the  "  modulus  "  ;  Cauchy  (1828)  called 
cos 0 +^ sin 0 the  "  reduced  form"(rexpression  reduite);  Gauss 
used  i  for  V  —  i,  introduced  the  term  "  complex  number  "  for 
a  -{-  H,  and  called  a^  +  b"  the  "  norm."  The  expression 
"  direction  coefficient  ",  often  used  for  cos  0  -f  /sin  0,  is  due 
to  Hankel  (1867),  and  "absolute  value,"  for  *' modulus,"  is 
due  to    Weierstrass. 

Following  Cauchy  and  Gauss  have  come  a  number  of  con- 
tributors of  high  rank,  of  whom  the  following  maybe  especially 
mentioned:  Kummer  (1844),  Kronecker(i845),  Schefifler  (1845, 
1851,  1880),  Bellavitis  (1835,  1852),  Peacock  (1845),  and  De 
Morgan  (1849).  Mobius  must  also  be  mentioned  for  his  num- 
erous memoirs  on  the  geometric  applications  of  complex 
numbers,   and  Dirichlet  for  the  expansion  of  the  theory  to  in- 


Art.  5.]  quaternions  and  ausdehnungslehre.  517 

elude  primes,  congruences,   reciprocity,  etc.,  as  in  the  case  of 
real  numbers. 

Other  types*  have  been  studied,  besides  the  famihar  a  -\-biy 
in  which  i  is  the  root  of  x'  -\-  i  =^  o.  Thus  Eisenstein  has 
studied  the  type  a  -\-  bj,j  being  a  complex  root  of  ;t:^  —  i  =  o. 
Similarly,  complex  types  have  been  derived  from  x''  —  i  z=zq 
(k  prime).  This  generalization  is  largely  due  to  Kummer,  to 
whom  is  also  due  the  theory  of  Ideal  Numbers, f  which  has 
recently  been  simplified  by  Klein  (1893)  from  the  point  of  view 
of  geometry.  A  further  complex  theory  is  due  to  Galois,  the 
basis  being  the  imaginary  roots  of  an  irreducible  congruence, 
F {x)^o  (mod/,  a  prime).  The  late  writers  (from  1884)  on 
the  general  theory  include  Weierstrass,  Schwarz,  Dedekind, 
Holder,  Berloty,  Poincard,  Study,  and  Macfarlane. 

Art.  5.    Quaternions  and  Ausdehnungslehre. 

Quaternions  and  Ausdehnungslehreif  are  so  closely  related 
to  complex  quantity,  and  the  latter  to  complex  number,  that 
the  brief  sketch  of  their  development  is  introduced  at  this 
point.  Caspar  Wessel's  contributions  to  the  theory  of  com- 
plex quantity  and  quaternions  remained  unnoticed  in  the 
proceedings  of  the  Copenhagen  Academy.  Argand's  attempts 
to  extend  his  method  of  complex  numbers  beyond  the  space 
of  two  dimensions  failed.  Servois  (1813),  however,  almost 
trespassed  on  the  quaternion  field.  Nevertheless  there  were 
fewer  traces  of  the  theory  anterior  to  the  labors  of  Hamilton 
than  is  usual  in  the  case  of  great  discoveries.  Hamilton  dis- 
covered the  principle  of  quaternions  in  1843,  ^"<^  ^^^  next  year 
his  first  contribution  to  the  theory  appeared,  thus  extending 
the  Argand  idea  to  three-dimensional  space.     This  step  neces- 

*  Chapman,  C.  H.,  Weierstrass  and  Dedekind  on  General  Complex  Num- 
bers, in  Bulletin  New  York  Mathematical  Society,  Vol.  I,  p.  150;  Study,  E., 
Aeltere  und  neuere  Untersuchungen  tiber  Systeme  complexer  Zahlen,  Mathe- 
matical Papers  Chicago  Congress,  p.  367;  bibliography,  p.  381. 

f  Klein,  F.,  Evanston  Lectures,  Lect.  VIII. 

X  Tait,  P.  G.,  on  Quaternions,  Encyclopaedia  Britannica;  Schlegel,  V.,  Die 
Grassmann'sche  Ausdehnungslehre,  SchlOmilch's  Zeitschrift,  Vol.  XLI, 


518  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XL 

sitated  an  expansion  of  the  idea  of  r(cos  0 +ysin  0)  such 
that  while  r  should  be  a  real  number  and  0  a  real  angle,  i,  j\ 
or  k  should  be  any  directed  unit  line  such  that  / "  —  j""  —  k"  —  —  i. 
It  also  necessitated  a  withdrawal  of  the  commutative  law  of 
multiplication,  the  adherence  to  which  obstructed  earlier  dis- 
covery. It  was  not  until  1853  that  Hamilton's  Lectures  on 
Quarternions  appeared,  followed  (1866)  by  his  Elements  of 
Quaternions. 

In  the  same  year  in  which  Hamilton  published  his  discov- 
ery (1844),  Grassmann  gave  to  the  world  his  famous  work, 
Die  lineale  Ausdehnungslehre,  although  he  seems  to  have 
been  in  possession  of  the  theory  as  early  as  1840.  Differing 
from  Hamilton's  Quaternions  in  many  features,  there  are 
several  essential  principles  held  in  common  which  each  writer 
discovered  independently  of  the  other. "^ 

Following  Hamilton,  there  have  appeared  in  Great  Britain 
numerous  papers  and  works  by  Tait  (1867),  Kelland  and  Tait 
(1873),  Sylveste;r,  and  McAulay  (1893).  On  the  Continent 
Hankel  (1867),  Hoiiel  (1874),  and  Laisant  (1877,  1881)  have 
written  on  the  theory,  but  it  has  attracted  relatively  little 
attention.  In  America,  Benjamin  Peirce  (1870)  has  been 
especially  prominent  in  developing  the  quaternion  theory,  and 
Hardy  (1881)  Macfarlane,  and  Hathaway  (1896)  have  con- 
tributed to  the  subject.  The  difficulties  have  been  largely  in 
the  notation.  In  attempting  to  improve  this  symbolism  Macfar- 
lane has  aimed  at  showing  how  a  space  analysis  can  be  de- 
veloped embracing  algebra,  trigonometry,  complex  numbers, 
Grassmann's  method,  and  quaternions,  and  has  considered  the 
general  principles,  of  vector  and  versor  analysis,  the  versor 
being  circular,  elHptic  logarithmic,  or  hyperbolic.  Other  recent 
contributors  to  the  algebra  of  vectors  are  Gibbs  (from  1881) 
and  Heaviside  (from  1885). 

The  followers  of  Grassmann  f  have  not  been  much  more 

I       *  These  are  set  forth  in  a  paper  by  J.  W.  Gibbs,  Nature,  Vol,  XLIV,  p.  79. 
f  For  bibliography   see  Schlegel,    V.,   Die  Grassmann'sche  Ausdehnungs- 
lehre, Schlomilch's  Zeitschrift,  Vol.  XLI. 


Art.  6.]  theory  of  equations.  519 

numerous  than  those  of  Hamilton.  Schlegel  has  been  one  of 
the  chief  contributors  in  Germany,  and  Peano  in  Italy.  In 
America,  Hyde  (Directional  Calculus,  1890)  has  made  a  plea 
for  the  Grassmann  theory.^ 

Along  lines  analogous  to  those  of  Hamilton  and  Grassmann 
have  been  the  contributions  of  Scheffler.  While  the  two 
former  sacrificed  the  commutative  law,  Scheffler  (1846,  185 1, 
1880)  sacrificed  the  distributive.  This  sacrifice  of  fundamental 
laws  has  led  to  an  investigation  of  the  field  in  which  these 
laws  are  valid,  an  investigation  to  which  Grassmann  (1872), 
Cayley,  Ellis,  Boole,  Schroder  (1890-91),  and  Kraft  (1893) 
have  contributed.  Another  great  contribution  of  Cayley's 
along  similar  lines  is  the  theory  of  matrices  (1858). 

Art.  6.    Theory  of  Equations. 

The  Theory  of  Numerical  Equations  f  concerns  itself  first 
with  the  location  of  the  roots,  and  then  with  their  approxi- 
mation. Neither  problem  is  new,  but  the  first  noteworthy 
contribution  to  the  former  in  the  nineteenth  century  was 
Sudan's  (1807).  Fourier's  work  was  undertaken  at  about  the 
same  time,  but  appeared  posthumously  in  1831.  All  processes 
were,  however,  exceedingly  cumbersome  until  Sturm  (1829) 
communicated  to  the  French  Academy  the  famous  theorem 
which  bears  his  name  and  which  constitutes  one  of  the  most 
brilliant  discoveries  of  algebraic  analysis. 

The  Approximation  of  the  Roots,  once  they  are  located, 
can  be  made  by  several  processes.  Newton  (i/i  i),  for  example, 
gave  a  method  which  Fourier  perfected;  and  Lagrange  (1767) 
discovered  an  ingenious  way  of  expressing  the  root  as  a  con- 
tinued fraction,  a  process  which  Vincent  (1836)  elaborated.     It 

*  For  Macfarlane's  Digest  of  views  of  English  and  American  writers,  see 
Proceedings  American  Association  for  Advancement  of  Science,  1891. 

f  Cayley,  A.,  Equations,  and  Kelland.  P.,  Algebra,  in  Encyclopaedia  Bri- 
tannica;  Favaro,  A.,  Notizie  storico-critiche  suUa  costruzione  delle  equazioni. 
Modena,  1878;  Cantor,  M.,  Geschichte  der  Mathematik,  Vol.  Ill,  p.  375. 


520  HISTORY    OF    MODERN    MATHEMATICS.  [ChAP.  XI. 

was,  however,  reserved  for  Horner  (1819)  to  suggest  the  most 
practical  method  yet  known,  the  one  now  commonly  used. 
With  Horner  and  Sturm  this  branch  practically  closes.  The 
calculation  of  the  imaginary  roots  by  approximation  is  still  an 
open  field. 

The  Fundamental  Theorem*  that  every  numerical  equation 
has  a  root  was  generally  assumed  until  the  latter  part  of  the 
eighteenth  century.  D'Alembert  (1746)  gave  a  demonstration, 
as  did  Lagrange  (1772),  Laplace  (1795),  Gauss  (1799)  and  Argand 
(1806).  The  general  theorem  that  every  algebraic  equation  of 
the  ;^th  degree  has  exactly  ?/  roots  and  no  more  follows  as  a 
special  case  of  Cauchy's  proposition  (1831)  as  to  the  number  of 
roots  within  a  given  contour.  Proofs  are  also  due  to  Gauss, 
Serret,  Clifford  (1876),  Malet  (1878),  and  many  others. 

The  Impossibility  of  Expressing  the  Roots  of  an  equation 
as  algebraic  functions  of  the  coefficients  when  the  degree  ex- 
ceeds 4  was  anticipated  by  Gauss  and  announced  by  Ruffini, 
and  the  belief  in  the  fact  became  strengthened  by  the  failure 
of  Lagrange's  methods  for  these  cases.  But  the  first  strict 
proof  is  due  to  Abel,  whose  early  death  cut  short  his  labors  in 
this  and  other  fields. 

The  Quintic  Equation  has  naturally  been  an  object  of 
special  study.  Lagrange  showed  that  its  solution  depends  on 
that  of  a  sextic,  *'  Lagrange's  resolvent  sextic,"  and  Malfatti 
and  Vandermonde  investigated  the  construction  of  resolvents. 
The  resolvent  sextic  was  somewhat  simplified  by  Cockle  and 
Harley  (1858-59)  and  by  Cayley  (1861),  but  Kronecker  (1858) 
was  the  first  to  establish  a  resolvent  by  which  a  real  simplifi- 
cation was  effected.  The  transformation  of  the  general  quintic 
into  the  trinomial  form  x" -\- ax -\- b  =  o  by  the  extraction  of 
square  and  cube  roots  only,  was  first  shown  to  be  possible  by 

*  Loria,  Gino,  Esame  di  alcune  ricerche  concernenti  Tesistenza  di  radici 
nelle  equazioni  algebriche;  Bibliotheca  Mathematica,  1891,  p.  99;  bibliography 
on  p.  107.  Pierpont,  J.,  On  the  Ruffini-Abelian  theorem,  Bulletin  of  American 
Mathematical  Society,  Vol.  II,  p.  200. 


^RT.  6.j  THEORY    OF    EQUATIONS.  521 

Bring  (1786)  and  independently  by  Jerrard  *  (1834).  Hermite 
(1858)  actually  effected  this  reduction,  by  means  of  Tschirn- 
hausen's  theorem,  in  connection  with  his  solution  by  elliptic 
functions. 

The  Modern  Theory  of  Equations  may  be  said  to  date  from 
Abel  and  Galois.  The  latter's  special  memoir  on  the  subject, 
not  pubHshed  until  1846,  fifteen  years  after  his  death,  placed 
the  theory  on  a  definite  base.  To  him  is  due  the  discovery 
that  to  each  equation  corresponds  a  group  of  substitutions 
(the  ''group  of  the  equation")  in  which  are  reflected  its  essen- 
tial characteristics.f  Galois's  untimely  death  left  without  suffi- 
cient demonstration  several  important  propositions,  a  gap 
^hich  Betti  (1852)  has  filled.  Jordan,  Hermite,  and  Kronecker 
were  also  among  the  earlier  ones  to  add  to  the  theory.  Just 
prior  to  Galois's  researches  Abel  (1824),  proceeding  from  the 
fact  that  a  rational  function  of  five  letters  having  less  than  five 
values  cannot  have  more  than  two,  showed  that  the  roots  of  a 
general  quintic  equation  cannot  be  expressed  in  terms  of 
its  coefficients  by  means  of  radicals.  He  then  investigated 
special  forms  of  quintic  equations  which  admit  of  solution  by 
the  extraction  of  a  finite  number  of  roots.  Hermite,  Sylves- 
ter, and  Brioschi  have  applied  the  invariant  theory  of  binary 
forms  to  the  same  subject. 

From  the  point  of  view  of  the  group  the  solution  by  radi- 
cals, formerly  the  goal  of  the  algebraist,  now  appears  as  a 
single  link  in  a  long  chain  of  questions  relative  to  the  transfor- 
mation of  irrationals  and  to  their  classification.  Klein  (1884) 
has  handled  the  whole  subject  of  the  quintic  equation  in  a 
simple  manner  by  introducing  the  icosahedron  equation  as  the 
normal  form,  and  has  shown  that  the  method  can  be  general- 
ized so  as  to  embrace  the  whole  theory  of  higher  equations.;]: 
He  and  Gordan  (from  1879)  have  attacked  those  equations  of 

*  Harley,   R.,  A  contribution  of  the  history  ...  of  the  general  equation 
-of  the  fifth  degree,  Quarterly  Journal  of  Mathematics,  Vol.  VI,  p.  38. 
f  See  Art.  7. 
X  Klein,  F.,  Vorlesungen  iiber  das  Ikosaeder,  1884. 


522  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XL 

the  sixth  and  seventh  degrees  which  have  a  Galois  group  of 
i68  substitutions,  Gordan  performing  the  reduction  of  the 
equation  of  the  seventh  degree  to  the  ternary  problem.  Klein 
(1888)  has  shown  that  the  equation  of  the  twenty-seventh 
degree  occurring  in  the  theory  of  cubic  surfaces  can  be  re- 
duced to  a  normal  problem  in  four  variables,  and  Burkhardt 
(1893)  has  performed  the  reduction,  the  quaternary  groups  in- 
volved having  been  discussed  by  Maschke  (from  1887). 

Thus  the  attempts  to  solve  the  quintic  equation  by  means 
of  radicals  has  given  place  to  their  treatment  by  transcendents. 
Hermite  (1858)  has  shown  the  possibility  of  the  solution,  by  the 
use  of  elliptic  functions,  of  any  Bring  quintic,  and  hence  of  any 
equation  of  the  fifth  degree.  Kronecker  (1858),  working  from 
a  different  standpoint,  has  reached  the  same  results,  and  his 
method  has  since  been  simplified  by  Brioschi.  More  recently 
Kronecker,  Gordan,  Kiepert,  and  Klein,  have  contributed  to 
the  same  subject,  and  the  sextic  equation  has  been  attacked  by 
Maschke  and  Brioschi  through  the  medium  of  hyperelliptic 
functions. 

Binomial    Equations,    reducible    to    the   form  x'^  —  i  =  o, 

admit  of  ready  solution  by  the  familiar  trigonometric  formula 

2k7t       ,       2kn 
X  =  cos  -j-^sm  —  ;  but  it  was  reserved  for  Gauss  (i8oiy 

to  show  that  an  algebraic  solution  is  possible.  Lagrange 
(1808)  extended  the  theory,  and  its  application  to  geometry  is^ 
one  of  the  leading  additions  of  the  century.  Abel,  generaliz- 
ing Gauss's  results,  contributed  the  important  theorem  that  if 
two  roots  of  an  irreducible  equation  are  so  connected  that  the 
one  can  be  expressed  rationally  in  terms  of  the  other,  the  equa- 
tion yields  to  radicals  if  the  degree  is  prime  and  otherwise 
depends  on  the  solution  of  lower  equations.       The    binomial 

M— I 

equation,  or  rather  the  equation   ^  x'""  =  o,  is  one  of  this  class. 

0 

considered  by  Abel,  and  hence  called  (by  Kronecker)  Abelian 

Equations.     The  binomial  equation  has  been  treated   notably 

by  Richelot  (1832),  Jacobi  (1837),  Eisenstein  (1844,  1850),  Cay- 


Art.  6. J  theory  of  equations.  523' 

ley  (185 1),  and  Kronecker  (1854),  and  is  the  subject  of  a 
treatise  by  Bachmann  (1872).  Among  the  most  recent  writers. 
on  AbeHan  equations  is  Pellet  (1891). 

Certain  special  equations  of  importance  in  geometry  have 
been  the  subject  of  study  by  Hesse,  Steiner,  Cayley,  Clebsch, 
Salmon,  and  Kummer.  Such  are  equations  of  the  ninth  degree 
determining  the  points  of  inflection  of  a  curve  of  the  third  de- 
gree, and  of  the  twenty-seventh  degree  determining  the  points 
in  which  a  curve  of  the  third  degree  can  have  contact  of  the 
fifth  order  with  a  conic. 

Symmetric  Functions  of  the  coefficients,  and  those  which  re- 
main unchanged  through  some  or  all  of  the  permutations  of  the 
roots,  are  subjects  of  great  importance  in  the  present  theory. 
The  first  formulas  for  the  computation  of  the  symmetric  func- 
tions of  the  roots  of  an  equation  seem  to  have  been  worked  out 
by  Newton,  although  Girard  (1629)  had  given,  without  proof,  a 
formula  for  the  power  sum.  In  the  eighteenth  century  Lagrange 
(1768)  and  Waring  (1770,  1782)  contributed  to  the  theory,  but 
the  first  tables,  reaching  to  the  tenth  degree,  appeared  in  1809 
in  the  Meyer- Hirsch  Aufgabensammlung.  In  Cauchy's  cele- 
brated memoir  on  determinants  (18 12)  the  subject  began  to 
assume  new  prominence,  and  both  he  and  Gauss  (1816)  made 
numerous  and  valuable  contributions  to  the  theory.  It  is,  how- 
ever, since  the  discoveries  by  Galois  that  the  subject  has  be- 
come one  of  great  importance.  Cayley  (1857)  has  given  sim- 
ple rules  for  the  degree  and  weight  of  symmetric  functions,  and 
he  and  Brioschi  have  simplified  the  computation  of  tables. 

Metltods  of  Elimination  and  of  finding  the  resultant 
(Bezout)  or  eliminant  (De  Morgan)  occupied  a  number  of 
eighteenth-century  algebraists,  prominent  among  them  being 
Euler  (1748),  whose  method,  based  on  symmetric  functions,  was 
improved  by  Cramer  (1750)  and  Bezout  (1764).  The  leading 
steps  in  the  development  are  represented  by  Lagrange  (1770-71),. 
Jacobi,  Sylvester  (1840),  Cayley  (1848,  1857),  Hesse  (1843,. 
1859),  Bruno  (1859),  ^"^  Katter  (1876).  Sylvester's  dialytfc 
method  appeared  in   1841,  and  to  him  is  also  due  (185 1)  the 


524  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XI. 

name  and  a  portion  of  the  theory  of  the  discriminant.  Among 
recent  writers  on  the  general  theory  may  be  mentioned  Burn- 
side  and  Pellet  (from  1887). 

Art.  7.    Substitutions  and  Groups. 

The  Theories  of  Substitutions  and  Groups*  are  among  the 
most  important  in  the  whole  mathematical  field,  the  study  of 
groups  and  the  search  for  invariants  now  occupying  the  atten- 
tion of  all  mathematicians.  The  first  recognition  of  the  im- 
portance of  the  combinatory  analysis  occurs  in  the  problem  of 
forming  an  wth-degree  equation  having  for  roots  ni  of  the  roots 
of  a  given  /^th-degree  equation  {in  <  n).  For  simple  cases 
the  problem  goes  back  to  Hudde  (1659).  Saunderson  (1740) 
noted  that  the  determination  of  the  quadratic  factors  of  a  bi- 
quadratic expression  necessarily  leads  to  a  sextic  equation,  and 
Le  Soeur  (1748)  and  Waring  (1762  to  1782)  still  further  elabo- 
rated the  idea. 

Lagrangef  first  undertook  a  scientific  treatment  of  the  the- 
ory of  substitutions.  Prior  to  his  time  the  various  methods  of 
solving  lower  equations  had  existed  rather  as  isolated  artifices 
than  as  a  unified  theory.;]:  Through  the  great  power  of  analy- 
sis possessed  by  Lagrange  (1770,  1771)  a  common  foundation 
was  discovered,  and  on  this  was  built  the  theory  of  substitu- 
tions. He  undertook  to  examine  the  methods  then  known, 
and  to  show  a  priori  why  these  succeeded  below  the  quintic, 
but  otherwise  failed.  In  his  investigation  he  discovered  the 
important  fact  that  the  roots  of  all  resolvents  (r^solvantes,  re- 
duites)  which  he  examined  are  rational  functions  of  the  roots 
of  the  respective  equations.  To  study  the  properties  of  these 
functions  he  invented  a  "  Calcul  des  Combinaisons,"  the  first 

*  Netto,  E.,  Theory  of  Substitutions,  translated  by  Cole;  Cay  ley,  A.,  Equa- 
tions, Encyclopaedia  Britannica,  9th  edition. 

f  Pierpont,  James,  Lagrange's  Place  in  the  Theory  of  Substitutions,  Bulletin 
of  American  Mathematical  Society,  Vol.  I,  p.  196. 

:J:Matthiessen,  L.,  Grundziige  der  antiken  und  modernen  Algebra  der  littera- 
len  Gleichungen,  Leipzig,  1878. 


Art.  7.]  substitutions  and  groups.  525 

important  step  towards  a  theory  of  substitutions.  Mention 
should  also  be  made  of  the  contemporary  labors  of  Vander- 
monde  (1770)  as  foreshadowing  the  coming  theory. 

The  next  great  step  was  taken  by  RufBni"^  (1799).  Begin- 
ning like  Lagrange  with  a  discussion  of  the  methods  of  solving 
lower  equations,  he  attempted  the  proof  of  the  impossibility  of 
solving  the  quintic  and  higher  equations.  While  the  attempt 
failed,  it  is  noteworthy  in  that  it  opens  with  the  classification 
of  the  various  "permutations"  of  the  coefficients,  using  the 
word  to  mean  what  Cauchy  calls  a  *'syst^me  des  substitutions 
conjuguees,"  or  simply  a  "  systeme  conjugu6,"  and  Galois  calls 
a  *'  group  of  substitutions."  Rufifini  distinguishes  what  are  now 
called  intransitive,  transitive  and  imprimitive,  and  transitive 
and  primitive  groups,  and  (1801)  freely  uses  the  group  of  an 
equation  under  the  name  'Tassieme  della  permutazioni."  He 
also  publishes  a  letter  from  Abbati  to  himself,  in  which  the 
group  idea  is  prominent. 

To  Galois,  however,  the  honor  of  establishing  the  theory  of 
groups  is  generally  awarded.  He  found  that  if  r^,  r^, .  .  .  r„  are 
the  n  roots  of  an  equation,  there  is  always  a  group  of  permuta- 
tions of  the  r's  such  that  (i)  every  function  of  the  roots  invari- 
able by  the  substitutions  of  the  group  is  rationally  known,  and 
(2),  reciprocally,  every  rationally  determinable  function  of  the 
roots  is  invariable  by  the  substitutions  of  the  group.  Galois 
also  contributed  to  the  theory  of  modular  equations  and  to  that 
of  elliptic  functions.  His  first  publication  on  the  group  theory 
was  made  at  the  age  of  eighteen  (1829),  but  his  contributions 
attracted  little  attention  until  the  publication  of  his  collected 
papers  in  1846  (Liouville,  Vol.  XI). 

Cayley  and  Cauchy  were  among  the  first  to  appreciate  the 
importance  of  the  theory,  and  to  the  latter  especially  are  due  a 
number  of  important  theorems.  The  popularizing  of  the  sub- 
ject is  largely  due  to  Serret,  who  has  devoted  section  IV  of  his 

♦Burkhardt,  H.,  Die  Anfange  der  Gruppentheorie  und  Paolo  Ruffini,  Ab- 
handlungen  zur  Geschichte  der  Mathematik,  VI,  1892,  p.  119.  Italian  by  E.  Pas- 
cal, Brioschi's  Annali  di  Matematici,  1894. 


526  HISTORY    OF    MODERN    MATHEMATICS.  [ChAP.  XI. 

algebra  to  the  theory ;  to  Camille  Jordan,  whose  Traite  des 
Substitutions  is  a  classic  ;  and  to  Netto  (1882),  whose  work  has 
been  translated  into  English  by  Cole  (1892).  Bertrand,  Her- 
mite,  Frobenius,  Kronecker,  and  Mathieu  have  added  to  the 
theory.  The  general  problem  to  determine  the  number  of 
groups  of  n  given  letters  still  awaits  solution. 

But  overshadowing  all  others  in  recent  years  in  carrying  on 
the  labors  of  Galois  and  his  followers  in  the  study  of  discontin- 
uous groups  stand  Klein,  Lie,  Poincare,  and  Picard.  Besides 
these  discontinuous  groups  there  are  other  classes,  one  of 
which,  that  of  finite  continuous  groups,  is  especially  important 
in  the  theory  of  differential  equations.  It  is  this  class  which 
Lie  (from  1884)  has  studied,  creating  the  most  important  of 
the  recent  departments  of  mathematics,  the  theory  of  trans- 
formation groups.  Of  value,  too,  have  been  the  labors  of 
Xilling  on  the  structure  of  groups,  Study's  application  of  the 
group  theory  to  complex  numbers,  and  the  work  of  Schur  and 
Mauren 

Art.  8.    Determinants. 

The  Theory  of  Determinants*  may  be  said  to  take  its 
origin  with  Leibniz  (1693),  following  whom  Cramer  (1750) 
added  slightly  to  the  theory,  treating  the  subject,  as  did  his 
predecessor,  wholly  in  relation  to  sets  of  equations.  The  re- 
current law  was  first  announced  by  Bezout  (1764).  But  it  was 
Vandermonde  (1771)  who  first  recognized  determinants  as  inde- 
pendent functions.  To  him  is  due  the  first  connected  exposi- 
tion of  the  theory,  and  he  may  be  called  its  formal  founder. 
Laplace  (1772)  gave  the  general  method  of  expanding  a  deter- 
minant in  terms  of  its  complementary  minors,  although  Van- 
dermonde had  already  given  a  special  case.  Immediately  fol- 
lowing, Lagrange   (1773)   treated    determinants  of  the  second 

*  Muir,  T.,  Theory  of  Determinantts  in  the  Historical  Order  of  its  Develop- 
ment, Part  I,  1890;  Baltzer,  R.,  Theorie  und  Anwendung  der  Determinanten, 
1881.  The  writer  is  under  obligations  to  Professor  Weld,  who  contributes 
Chap.  II,  for  valuable  assistance  in  compiling  this  article. 


Art.  b.J  DETERMINANTS.  537 

and  third  order,  possibly  stopping  here  because  the  idea  of 
hyperspace  was  not  then  in  vogue.  Although  contributing 
nothing  to  the  general  theory,  Lagrange  was  the  first  to  apply 
determinants  to  questions  foreign  to  eliminations,  and  to  him 
are  due  many  special  identities  which  have  since  been  brought 
under  well-known  theorems.  During  the  next  quarter  of  a 
century  little  of  importance  was  done.  Hindenburg  (1784)  and 
Rothe  (1800)  kept  the  subject  open,  but  Gauss  (1801)  made 
the  next  advance.  Like  Lagrange,  he  made  much  use  of  de- 
terminants in  the  theory  of  numbers.  He  introduced  the  word 
*' determinants  "  (Laplace  had  used  "  resultant  "),  though  not 
in  the  present  signification,*  but  rather  as  applied  to  the  dis- 
criminant of  a  quantic.  Gauss  also  arrived  at  the  notion  of 
reciprocal  determinants,  and  came  very  near  the  multiplication 
theorem.  The  next  contributor  of  importance  is  Binet  (181 1, 
1812),  who  formally  stated  the  theorem  relating  to  the  product 
of  two  matrices  of  m  columns  and  n  rows,  which  for  the  special 
case  of  in  =  n  reduces  to  the  multiplication  theorem.  On  the 
same  day  (Nov.  30,  1812)  that  Binet  presented  his  paper  to  the 
Academy,  Cauchy  also  presented  one  on  the  subject.  In  this 
he  used  the  word  "  determinant "  in  its  present  sense,  summa- 
rized and  simplified  what  was  then  known  on  the  subject,  im- 
proved the  notation,  and  gave  the  multiplication  theorem  with 
a  proof  more  satisfactory  than  Binet's.  He  was  the  first  to 
grasp  the  subject- as  a  whole;  before  him  there  were  determi- 
nants, with  him  begins  their  theory  in  its  generality. 

The  next  great  contributor,  and  the.  greatest  save  Cauchy, 
was  Jacobi  (from  1827).  With  him  the  word  "determinant" 
received  its  final  acceptance.  He  early  used  the  functional 
determinant  which  Sylvester  has  called  the  "  Jacobian,"  and  in 
his  famous  memoirs  in  Crelle  for  1841  he  specially  treats  this 
subject,  as  well  as  that  class  of  alternating  functions  which 
Sylvester  has  called  ''Alternants."  But  about  the  time  of 
Jacobi's  closing  memoirs,  Sylvester  (1839)  '^"^  Cayley  began 

*  "  Numerum  bb-ac,  cuius  indole  proprietates  formae  {a,  b,  c)  imprimis  pen- 
dcre  in  sequer;tl^n<?  docebimus,  determinantem  huius  uocabimus." 


528  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XI. 

their  great  work,  a  work  which  it  is  impossible  to  briefly  sum- 
marize, but  which  represents  the  development  of  the  theory  to 
the  present  time. 

The  study  of  special  forms  of  determinants  has  been  the 
natural  result  of  the  completion  of  the  general  theory.  Axi- 
symmetric  determinants  have  been  studied  by  Lebesgue,  Hesse, 
and  Sylvester ;  per-symmetric  determinants  by  Sylvester  and 
Hankel ;  circulants  by  Catalan,  Spottiswoode,  Glaisher,  and 
Scott;  skew  determinants  and  Pfaffians,  in  connection  with  the 
theory  of  orthogonal  transformation,  by  Cayley ;  continuants 
by  Sylvester;  Wronskians  (so  called  by  Muir)  by  Christoffel 
and  Frobenius  ;  compound  determinants  by  Sylvester,  Reiss^ 
and  Picquet ;  Jacobians  and  Hessians  by  Sylvester  ;  and  sym- 
metric gauche  determinants  by  Trudi.  Of  the  text-books  on 
the  subject  Spottiswoode's  was  the  first.  In  America,  Hanus 
(1886)  and  Weld  (1893)  have  published  treatises. 

Art.  9.    Quantics. 

The  Theory  of  Quantics  or  Forms  ^  appeared  in  embryo  in 
the  Berlin  memoirs  of  Lagrange  (1773,  1775),  who  considered 
binary  quadratic  forms  of  the  type  a:v^  -{-bxy  -|-  cy^,  and  estab- 
lished the  invariance  of  the  discriminant  of  that  type  when 
X -\- Xj/  is  put  for  X.  He  classified  forms  of  that  type  accord- 
ing to  the  sign  of  d^  —  4ac,  and  introduced  the  ideas  of  trans- 
formation and  equivalence.  Gauss  f  (1801)  next  took  up  the 
subject,  proved  the  invariance  of  the  discriminants  of  binary 
and  ternary  quadratic  forms,  and  systematized  the  theory  of 
binary  quadratic  forms,  a  subject  elaborated  by  H.  J.  S^ 
Smith,  Eisenstein,  Dirichlet,  Lipschitz,  Poincare,  and  Caylew 
Galois  also  entered  the  field,  in  his  theory  of  groups  (1829),  and 

*  Meyer,  W.  F.,  Bericht  Uber  den  gegenwanigen  Stand  der  Invarianten- 
theorie.  Jahresbericht  der  deutschen  Mathematiker-Vereinigung,  Vol.  I, 
1890-91;  Berlin  1892,  p.  97.  See  also  the  review  by  Franklin  in  Bulletin  New- 
York  Mathematical  Society,  Vol.  Ill,  p.  187  ;  Biography  of  Cayley,  Collected 
Papers,  VIII,  p.  ix,  and  Proceedings  of  Royal  Society,  1895. 

f  See  Art.  2. 


Art.  9.]  quantics.  529- 

the  first  step  towards  the  establishment  of  the  distinct  theory- 
is  sometimes  attributed  to  Hesse  in  his  investigations  of  the 
plane  curve  of  the  third  order. 

It  is,  however,  to  Boole  (1841)  that  the  real  foundation  of 
the  theory  of  invariants  is  generally  ascribed.  He  first  showed 
the  generality  of  the  invariant  property  of  the  discriminant, 
which  Lagrange  and  Gauss  had  found  for  special  forms. 
Inspired  by  Boole's  discovery  Cayley  took  up  the  study  in  a 
memoir  "On  the  Theory  of  Linear  Transformations  "  (1845), 
which  was  followed  (1846)  by  investigations  concerning  co- 
variants  and  by  the  discovery  of  the  symbolic  method  of  find- 
ing invariants.  By  reason  of  these  discoveries  concerning 
invariants  and  covariants  (which  at  first  he  called  "  hyperdeter- 
minants  ")  he  is  regarded  as  the  founder  of  what  is  variously 
called  Modern  Algebra,  Theory  of  Forms,  Theory  of  Quantics, 
and  the  Theory  of  Invariants  and  Covariants.  His  ten  memoirs 
on  the  subject  began  in  1854,  and  rank  among  the  greatest 
which  have  ever  been  produced  upon  a  single  theory.  Syl- 
vester soon  joined  Cayley  in  this  work,  and  his  originality  and 
vigor  in  discovery  soon  made  both  himself  and  the  subject 
prominent.  To  him  are  due  (1851-54)  the  foundations  of  the 
general  theory,  upon  which  later  writers  have  largely  built,  as 
well  as  most  of  the  terminology  of  the  subject. 

Meanwhile  in  Germany  Eisenstein  (1843)  had  become  aware 
of  the  simplest  invariants  and  covariants  of  a  cubic  and  bi- 
quadratic form,  and  Hesse  and  Grassmann  had  both  (1844) 
touched  upon  the  subject.  But  it  was  Aronhold  (1849)  who 
first  made  the  new  theory  known.  He  devised  the  symbolic 
method  now  common  in  Germany,  discovered  the  invariants 
of  a  ternary  cubic  and  their  relations  to  the  discriminant,  and, 
with  Cayley  and  Sylvester,  studied  those  differential  equations 
which  are  satisfied  by  invariants  and  covariants  of  binary 
quantics.  His  symbolic  method  has  been  carried  on  by 
Clebsch,  Gordan,  and  more  recently  by  Study  (1889)  and  Stroh 
(1890),  in  lines  quite  different  from  those  of  the  English  scbooL 

In  France  Hermite   early  took  up  the  work  (185 1).     He 


530  HISTORY    OF    MODERN    MATHEMATICS.  [ChAP.  XI. 

discovered  (1854)  the  law  of  reciprocity  that  to  every  covariant 
or  invariant  of  degree  p  and  order  r  of  a  form  of  the  mth 
order  corresponds  also  a  covariant  or  invariant  of  degree  m 
and  of  order  r  of  a  form  of  the  pth  order.  At  the  same  time 
(1854)  Brioschi  joined  the  movement,  and  his  contributions 
have  been  among  the  most  valuable.  Salmon's  Higher  Plane 
Curves  (1852)  and  Higher  Algebra  (1859)  should  also  be  men- 
tioned as  marking  an  epoch  in  the  theory. 

Gordan  entered  the  field,  as  a  critic  of  Cayley,  in  1868.  He 
added  greatly  to  the  theory,  especially  by  his  theorem  on  the 
Endlichkeit  des  Formensystems,  the  proof  for  which  has  since 
been  simpHfied.  This  theory  of  the  finiteness  of  the  number 
of  invariants  and  covariants  of  a  binary  form  has  since  been 
extended  by  Peano  (1882),  Hilbert  (1884),  and  Mertens  (1886). 
Hilbert  (1890)  succeeded  in  showing  the  finiteness  of  the  com- 
plete systems  for  forms  in  n  variables,  a  proof  which  Story  has 
simplified. 

Clebsch  *  did  more  than  any  other  to  introduce  into  Ger- 
many the  work  of  Cayley  and  Sylvester,  interpreting  the  pro- 
jective geometry  by  their  theory  of  invariants,  and  correlating 
it  with  Riemann's  theory  of  functions.  Especially  since  the 
publication  of  his  work  on  forms  (1871)  the  subject  has  at- 
tracted such  scholars  as  Weierstrass,  Kronecker,  Mansion, 
Noether,  Hilbert,  Klein,  Lie,  Beltrami,  Burkhardt,  and  many 
others.  On  binary  forms  Fak  di  Bruno's  work  is  well  known, 
as  is  Study's  (1889)  on  ternary  forms.  De  Toledo  (1889)  and 
Elliott  (1895)  have  published  treatises  on  the  subject. 

DubHn  University  has  also  furnished  a  considerable  corps 
of  contributors,  among  whom  MacCullagh,  Hamilton,  Salmon, 
Michael  and  Ralph  Roberts,  and  Burnside  may  be  especially 
mentioned.  Burnside,  who  wrote  the  latter  part  of  Burnside 
and  Panton's  Theory  of  Equations,  has  set  forth  a  method  of 
transformation  which  is  fertile  in  geometric  interpretation  and 
binds  together  binary  and  certain  ternary  forms. 

*  Klein's  Evanston  Lectures,  Lect.  I. 


Art.  10.]  CALCULUS.  531 

The  equivalence  problem  of  quadratic  and  bilinear  forms 
has  attracted  the  attention  of  Weierstrass,  Kronecker,  Chris- 
toffel,  Frobenius,  Lie,  and  more  recently  of  Rosenow  (Crelle, 
io8),  Werner  (1889),  KiUing  (1890),  and  Scheffers  (1891).  The 
equivalence  problem  of  non-quadratic  forms  has  been  studied 
by  Christoffel.  Schwarz  (1872),  Fuchs  (1875-76),  Klein  (1877, 
1884),  Brioschi  (1877),  and  Maschke  (1887)  have  contributed 
to  the  theory  of  forms  with  linear  transformations  into  them- 
selves. Cayley  (especially  from  1870)  and  Sylvester  (1877) 
have  worked  out  the  methods  of  denumeration  by  means  of 
generating  functions.  Differential  invariants  have  been  studied 
by  Sylvester,  MacMahon,  and  Hammond.  Starting  from  the 
differential  invariant,  which  Cayley  has  termed  the  Schwarzian 
derivative,  Sylvester  (1885)  has  founded  the  theory  of  recipro- 
cants,  to  which  MacMahon,  Hammond,  Leudesdorf,  Elliott, 
Forsyth,  and  Halphen  have  contributed.  Canonical  forms  have 
been  studied  by  Sylvester  (1851),  Cayley,  and  Hermite  (to 
whom  the  term  "  canonical  form  "  is  due),  and  more  recently 
by  Rosanes  (1873),  Brill  (1882J,  Gundelfinger  (1883),  and  Hil- 
bert  (1886). 

The  Geometric  Theory  of  Binary  Forms  may  be  traced  to 
Poncelet  and  his  followers.  But  the  modern  treatment  has  its 
origin  in  connection  with  the  theory  of  elliptic  modular  func- 
tions, and  dates  from  Dedekind's  letter  to  Borchardt  (Crelle, 
1877).  The  names  of  Klein  and  Hurwitz  are  prominent  in 
this  connection.  On  the  method  of  nets  (reseaux),  another 
geometric  treatment  of  binary  quadratic  forms  Gauss  (1831), 
Dirichlet  (1850),  and  Poincar^  (1880)  have  written. 

Art.  10.    Calculus. 

The  Differential  and  Integral  Calculus,*  dating  from  New- 
ton and  Leibniz,  was  quite  complete  in  its  general  range  at 

*  Williamson,  B.,  Infinitesimal  Calculus,  Encyclopaedia  Britannica,  9th  edi- 
tion; Cantor,  M.,  Geschichte  der  Mathematik,  Vol.  Ill,  pp.  150-316;  Vivanti,  G., 
Note  sur  I'histoire  de  Tinfiniment  petit,  Bibliotheca  Mathematica,  1894,  p.  i  ; 
Mansion,  P.,  Esquisse  de  Thistoire  du  calcul  infinitesimal,  Ghent,  1887.     Le 


532  HISTORY    OF    MODERN    MATHEMATICS.  [ChAP,  XL 

the  close  of  the  eighteenth  century.  Aside  from  the  study  of 
first  principles,  to  which  Gauss,  Cauchy,  Jordan,  Picard,  Meray, 
»  and  those  whose  names  are  mentioned  in  connection  with  the 
theory  of  functions,  have  contributed,  there  must  be  men- 
tioned the  development  of  symbolic  methods,  the  theory  of 
definite  integrals,  the  calculus  of  variations,  the  theory  of  dif- 
ferential equations,  and  the  numerous  applications  of  the 
Newtonian  calculus  to  physical  problems.  Among  those  who 
have  prepared  noteworthy  general  treatises  are  Cauchy  (1821), 
Raabe  (1839-47),  Duhamel  (1856),  Sturm  (1857-59),  Bertrand 
(1864),  Serret  (1868),  Jordan  (2d  ed.,  1893),  and  Picard  (1891-93). 
A  recent  contribution  to  analysis  which  promises  to  be  valu- 
able is  Oltramare's  Calcul  de  Generalization  (1893). 

Abel  seems  to  have  been  the  first  to  consider  in  a  general 
way  the  question  as  to  what  differential  expressions  can  be 
integrated  in  a  finite  form  by  the  aid  of  ordinary  functions,  an 
investigation  extended  by  Liouville.  Cauchy  early  undertook 
the  general  theory  of  determining  definite  integrals,  and  the 
subject  has  been  prominent  during  the  century.  Frullani's 
theorem  (1821),  Bierens  de  Haan's  work  on  the  theory  (1862) 
and  his  elaborate  tables  (1867),  Dirichlet's  lectures  (1858)  em- 
bodied in  Meyer's  treatise  (1871),  and  numerous  memoirs  of 
Legendre,  Poisson,  Plana,  Raabe,  Sohncke,  Schlomilch,  Elliott, 
Leudesdorf,  and  Kronecker  are  among  the  noteworthy  con- 
tributions. 

Eulerian  Integrals  were  first  studied  by  Euler  and  after- 
wards  investigated  by  Legendre,  by  whom  they  were  classed  as 
Eulerian  integrals  of  the  first  and  second  species,  as  follows : 

J    x''~\i  —  xf-^dx,  J  e-'^x'^-^dx,  although  these  were  not  the 
exact  forms  of  Euler's  study.     If  n  is  integral,  it  follows  that 

/  e'^'x^'^dx  =  n\,  but  if   n   is  fractional  it  is  a  transcendent 
function.     To  it  Legendre  assigned  the  symbol  F,  and  it  is 

deux  centi^me  anniversaire  de  I'invention   du  calcul  diff6rentiel  ;   Mathesis» 
Vol.  IV,  p.  163. 


Art.  10.]  CALCULUS.  533 

now  called  the  gamma  function.  To  the  subject  Dirichlet  has 
contributed  an  important  theorem  (Liouville,  1839),  which  has 
been  elaborated  by  Liouville,  Catalan,  Leslie  Ellis,  and  others. 
On  the  evaluation  of  Fx  and  log  Fx  Raabe  (1843-44),  Bauer 
(1859),  ^^^^  Gudermann  (1845)  have  written.  Legendre's  great 
table  appeared  in  1 8 16. 

Symbolic  Methods  may  be  traced  back  to  Taylor,  and  the 
analogy  between  successive  differentiation  and  ordinary  ex- 
ponentials had  been  observed  by  numerous  writers  before  the 
nineteenth  century.  Arbogast  (1800)  was  the  first,  however, 
to  separate  the  symbol  of  operation  from  that  of  quantity  in  a 
differential  equation.  Francois  (1812)  and  Servois  (18 14)  seem 
to  have  been  the  first  to  give  correct  rules  on  the  subject. 
Hargreave  (1848)  applied  these  methods  in  his  memoir  on  dif- 
ferential equations,  and  Boole  freely  employed  them.  Grass- 
mann  and  Hankel  made  great  use  of  the  theory,  the  former  in 
studying  equations,  the  latter  in  his  theory  of  complex  num- 
bers. 

The  Calculus  of  Variations^  may  be  said  to  begin  with  a 
problem  of  Johann  Bernoulli's  (1696).  It  immediately  occu- 
pied the  attention  of  Jakob  Bernoulli  and  the  Marquis  de 
THopital,  but  Euler  first  elaborated  the  subject.  His  contri- 
butions began  in  1733,  and  his  Elementa  Calculi  Variationum 
gave  to  the  science  its  name.  Lagrange  contributed  extensively 
to  the  theory,  and  Legendre  (1786)  laid  down  a  method, 
not  entirely  satisfactory,  for  the  discrimination  of  max- 
ima and  minima.  To  this  discrimination  Brunacci  (18 10), 
Gauss  (1829),  Poisson  (1831),  Ostrogradsky  (1834),  and  Jacobi 
(1837)  have  been  among  the  contributors.  An  important  gen- 
eral work  is  that  of  Sarrus  (1842)  which  was  condensed  and  im- 
proved by  Cauchy  (1844).  Other  valuable  treatises  and  me- 
moirs have  been  written  by  Strauch  (1849"),  Jellett  (1850),  Hesse 
(1857),  Clebsch  (1858),  and  Carll  (1885),  but  perhaps  the  most 

*  Carll,  L.  B.,  Calculus  of  Variations,  New  York,  1885,  Chap.  V;  Tod- 
hunter,  I.,  History  of  the  Progress  of  the  Calculus  of  Variations,  London, 
1 861  :  Reiff,  R.,  Die  AnfSnge  der  Variationsrechnung,  Mathematisch-natur- 
Tvissenschaftliche  Mittheilungen,  Tiibingen,  1887,  p.  90. 


534  HISTORY    VF   MODERN    MATHEMATICS.  [ChaP.  XL 

important  work  of  the  century  is  that  of  Weierstrass.  His 
celebrated  course  on  the  theory  is  epoch-making,  and  it  may 
be  asserted  that  he  was  the  first  to  place  it  on  a  firm  and  un- 
questionable foundation. 

The  Application  of  the  Infinitesimal  Calculus  to  problems 
in  physics  and  astronomy  was  contemporary  with  the  origin  of 
the  science.  All  through  the  eighteenth  century  these  appli- 
cations were  multiplied,  until  at  its  close  Laplace  and  Lagrange 
had  brought  the  whole  range  of  the  study  of  forces  into  the 
realm  of  analysis.  To  Lagrange  (1773)  we  owe  the  introduc- 
tion of  the  theory  of  the  potential*  into  dynamics,  although 
the  name  "potential  function"  and  the  fundamental  memoir 
of  the  subject  are  due  to  Green  (1827,  printed  in  1828).  The 
name  "potential"  is  due  to  Gauss  (1840),  and  the  distinction 
between  potential  and  potential  function  to  Clausius.  With 
its  development  are  connected  the  names  of  Dirichlet,  Rie- 
mann,  Neumann,  Heine,  Kronecker,  Lipschitz,  Christoffel, 
Kirchhoff,  Beltrami,  and  many  of  the  leading  physicists  of  the 
century. 

It  is  impossible  in  this  place  to  enter  into  the  great  variety 
of  other  apphcations  of  analysis  to  physical  problems.  Among 
them  are  the  investigations  of  Euler  on  vibrating  chords; 
Sophie  Germain  on  elastic  membranes ;  Poisson,  Lame,  Saint- 
Venant,  and  Clebsch  on  the  elasticity  of  three-dimensional  bod- 
ies; Fourier  on  heat  diffusion;  Fresnel  on  Hght;  Maxwell,  Helm- 
holtz,  and  Hertz  on  electricity;  Hansen,  Hill,  and  Gylden  on 
astronomy;  Maxwell  on  spherical  harmonics;  Lord  Rayleigh  on 
acoustics;  and  the  contributions  of  Dirichlet,  Weber,  Kirchhoff, 
F.  Neumann,  Lord  Kelvin,  Clausius,  Bjerknes,  MacCullagh, 
and  Fuhrmann  to  physics  in  general.  The  labors  of  Helm- 
holtz  should  be  especially  mentioned,  since  he  contributed  to 
the  theories  of  dynamics,  electricity,  etc.,  and  brought  his  great 
analytical  powers  to  bear  on  the  fundamental  axioms  of  me- 
chanics as  well  as  on  those  of  pure  mathematics. 

*  Bacharach,  M.,  Abriss  der  Geschichte  der  Potentialtheorie,  1883.  This- 
contains  an  extensive  bibliography. 


Art.  11.]  DIFFERENTIAL    EQUATIONS.  535 

Art.  11.     Differential  Equations. 

The  Theory  of  Differential  Equations  "^  has  been  called  by 
Lief  the  most  important  of  modern  mathematics.  The  influ- 
ence of  geometry,  physics,  and  astronomy,  starting  with  New- 
ton and  Leibniz,  and  further  manifested  through  the  BernouUis, 
Riccati,  and  Clairaut,  but  chiefly  through  d'Alcmbert  and 
Euler,  has  been  very  marked,  and  especially  on  the  theory  of 
linear  partial  differential  equations  with  constant  coefficients. 
The  first  method  of  integrating  linear  ordinary  differential 
equations  with  constant  coefficients  is  due  to  Euler,  who  made 

the  solution  of  his  type,  ,  „  +  ^i  /  ^-i  +  •  •  •  +  ^«  J  =  o,  de- 
pend on  that  of  the  algebraic  equation  of  the  ;/th  degree, 
F{£)  —  z'^-^A^z'''^-^-. .  .-\-A„  =  o,  in  which  z''  takes  the  place  of 

-J— fe  (k  =  I,  2y  .  .  .  n).  This  equation  F(z)  =  o,  is  the  "  char- 
acteristic "  equation  considered  later  by  Monge  and  Cauchy. 

The  theory  of  linear  partial  differential  equations  may  be 
said  to  begin  with  Lagrange  (1779  ^^  17^5)-  Monge  (1809) 
treated  ordinary  and  partial  differential  equations  of  the  first 
and  second  order,  uniting  the  theory  to  geometry,  and  intro- 
ducing the  notion  of  the  "  characteristic,"  the  curve  represented 
by  F{z)  =  o,  which  has  recently  been  investigated  by  Darboux, 

*  Cantor,  M.,  Geschichte  der  Mathematik,  Vol.  Ill,  p.  429  ;  Schlesinger,  L., 
Handbuch  der  Theorie  der  linearen  Differentialgleichungen,  Vol.  I,  1895,  an  ex- 
cellent historical  view  ;  review  by  Mathews  in  Nature,  Vol.  LII,  p.  313;  Lie,  S., 
Zur  allgemeinen  Theorie  der  paniellen  Differentialgleichungen,  Berichte  uber 
die  Verhandlungen  der  Gesellschaft  der  Wissenschaften  zu  Leipzig,  1895;; 
Mansion,  P.,  Theorie  der  partiellen  Differentialgleichungen  ler  Ordnung,  Ger- 
man by  Maser,  Leipzig,  1892,  excellent  on  history  ;  Craig,  T.,  Some  of  the  De- 
velopments in  the  Theory  of  Ordinary  Differential  Equations,  1878-1893,  Bul- 
letin New  York  Mathematical  Society,  Vol.  II,  p.  119  ;  Goursat,  E.,  Le9onssur 
rint6gration  des  Equations  aux  deriv^es  partielles  du  premier  ordre,  Paris,  1891; 
Burkhardt,  H.,  and  Hefter,  L.,  in  Mathematical  Papers  of  Chicago  Congress, 
p.  13  and  p.  96. 

f  "In  der  ganzen  modernen^  Mathematik  ist  die  Theorie  der  Differential- 
gleichungen die  wichiigste  Discipiin  " 


536  HISTORY    OF   MODERN    MATHEMATICS.  [Chap.  XL 

Levy,  and  Lie.  Pfaff  (1814,  18 15)  gave  the  first  general  method 
of  integrating  partial  differential  equations  of  the  first  order,  a 
method  of  which  Gauss  (181 5)  at  once  recognized  the  value 
and  of  which  he  gave  an  analysis.  Soon  after,  Cauchy  (18 19) 
gave  a  simpler  method,  attacking  the  subject  from  the  analyt- 
ical standpoint,  but  using  the  Monge  characteristic.  To  him 
is  also  due  the  theorem,  corresponding  to  the  fundamental 
theorem  of  algebra,  that  every  differential  equation  defines  a 
function  expressible  by  means  of  a  convergent  series,  a  propo- 
sition more  simply  proved  by  Briot  and  Bouquet,  and  also  by 
Picard  (1891).  Jacobi  (1827)  also  gave  an  analysis  of  Pfaff's 
method,  besides  developing  an  original  one  (1836)  which 
Clebsch  published  (1862).  Clebsch's  own  method  appeared  in 
1866,  and  others  are  due  to  Boole  (1859),  Korkine  (1869),  and 
A.  Mayer  (1872).  Pfaff's  problem  has  been  a  prominent  sub- 
ject of  investigation,  and  with  it  are  connected  the  names  of 
Natani  (1859),  Clebsch  (1861,  1862),  DuBois-Reymond  (1869), 
Cayley,  Baltzer,  Frobenius,  Morera,  Darboux,  and  Lie.  The 
next  great  improvement  in  the  theory  of  partial  differential 
equations  of  the  first  order  is  due  to  Lie  (1872),  by  whom  the 
whole  subject  has  been  placed  on  a  rigid  foundation.  Since 
about  1870,  Darboux,  Kovalevsky,  Meray,  Mansion,  Grain- 
dorge,  and  Imschenetsky  have  been  prominent  in  this  line. 
The  theory  of  partial  differential  equations  of  the  second  and 
higher  orders,  beginning  with  Laplace  and  Monge,  was  notably 
advanced  by  Ampere  (1840).  Imschenetsky  *  has  summarized 
the  contributions  to  1873,  but  the  theory  remains  in  an 
imperfect  state. 

The  integration  of  partial  differential  equations  with  three 
or  more  variables  was  the  object  of  elaborate  investigations  by 
Lagrange,  and  his  name  is  still  connected  with  certain  subsid- 
iary equations.  To  him  and  to  Charpit,  who  did  much  to 
develop  the  theory,  is  due  one  of  the  methods  for  integrating 
the  general  equation  with  two  variables,  a  method  which  now 
bears  Charpit's  name. 

*  Grunert's  Archiv  fur  Mathematik,  Vol.  LIV. 


Art.  11.]  DIFFERENTIAL    EQUATIONS.  537 

The  theory  of  singular  solutions  of  ordinary  and  partial 
differential  equations  has  been  a  subject  of  research  from  the 
time  of  Leibniz,  but  only  since  the  middle  of  the  present  cen- 
tury has  it  received  especial  attention.  A  valuable  but  little- 
known  work  on  the  subject  is  that  of  Houtain  (1854).  Dar- 
boux  (from  1873)  has  been  a  leader  in  the  theory,  and  in  the 
geometric  interpretation  of  these  solutions  he  has  opened  a 
field  which  has  been  worked  by  various  writers,  notably  Caso- 
rati  and  Cayley.  To  the  latter  is  due  (1872)  the  theory  of 
singular  solutions  of  differential  equations  of  the  first  order  as 
at  present  accepted. 

The  primitive  attempt  in  dealing  with  differential  equations 
had  in  view  a  reduction  to  quadratures.  As  it  had  been  the  hope 
of  eighteenth-century  algebraists  to  find  a  method  for  solving 
the  general  equation  of  the  nth  degree,  so  it  was  the  hope  of 
analysts  to  find  a  general  method  for  integrating  any  differen- 
tial equation.  Gauss  (1799)  showed,  however,  that  the  dif- 
ferential equation  meets  its  limitations  very  soon  unless 
complex  numbers  are  introduced.  Hence  analysts  began  to 
study  these  equations  as  functions,  thus  opening  a  new  and  fer- 
tile field.  Cauchy  was  the  first  to  appreciate  tiie  importance 
of  this  view,  and  the  modern  theory  may  be  said  to  begin  with 
him.  Thereafter  the  real  question  was  to  be,  not  whether  a 
solution  is  possible  by  means  of  known  functions  or  their  in- 
tegrals, but  whether  a  given  differential  equation  suffices  for 
the  definition  of  a  function  of  the  independent  variable  or 
variables,  and  if  so,  what  are  the  characteristic  properties  of 
this  function. 

Within  a  half-century  the  theory  of  ordinary  differential 
equations  has  come  to  be  one  of  the  most  important  branches 
of  analysis,  the  theory  of  partial  differential  equations  remain- 
ing as  one  still  to  be  perfected.  The  difficulties  of  the  general 
problem  of  integration  are  so  manifest  that  all  classes  of  inves- 
tigators have  confined  themselves  to  the  properties  of  the  in- 
tegrals in  the  neighborhood  of  certain  given  points.  The  new 
<ieparture  took  its  greatest  inspiration  from  two  memoirs  by 


538  HISTORY    OF    MODERN    MATHEMATICS.  [ChAP.  XI. 

Fuchs  (Crelle,  1866,  1868),  a  work  elaborated  by  Thome  and 
Frobenius.  Collet  has  been  a  prominent  contributor  since 
1869,  although  his  method  for  integrating  a  non-linear  system 
was  communicated  to  Bertrand  in  1868.  Clebsch  *  (1873)  at- 
tacked the  theory  along  lines  parallel  to  those  followed  in  his 
theory  of  Abelian  integrals.  As  the  latter  can  be  classified 
according  to  the  properties  of  the  fundamental  curve  which 
remains  unchanged  under  a  rational  transformation,  soClebsch 
proposed  to  classify  the  transcendent  functions  defined  by  the 
differential  equations  according  to  the  invariant  properties  of 
the  corresponding  surfaces  /=  o  under  rational  one-to-one 
transformations. 

Since  1870  Lie's  f  labors  have  put  the  entire  theory  of  dif- 
ferential equations  on  a  more  satisfactory  foundation.  He  has 
shown  that  the  integration  theories  of  the  older  mathema- 
ticians, which  had  been  looked  upon  as  isolated,  can  by  the 
introduction  of  the  concept  of  continuous  groups  of  transfor- 
mations be  referred  to  a  common  source,  and  that  ordinary 
differential  equations  which  admit  the  same  infinitesimal  trans- 
formations present  like  difificulties  of  integration.  He  has  alsa 
emphasized  the  subject  of  transformations  of  contact  (Beriih- 
rungstransformationen)  which  underlies  so  much  of  the  recent 
theory.  The  modern  school  has  also  turned  its  attention  to 
the  theory  of  differential  invariants,  one  of  fundamental  im- 
portance and  one  which  Lie  has  made  prominent.  With  this 
theory  are  associated  the  names  of  Cayley,  Cockle,  Sylvester, 
Forsyth,  Laguerre,  and  Halphen.  Recent  writers  have  shown 
the  same  tendency  noticeable  in  the  work  of  Monge  and 
Cauchy,  the  tendency  to  separate  into  two  schools,  the  one 
inclining  to  use  the  geometric  diagram,  and  represented  by 
Schwarz,  Klein,  and  Goursat,  the  other  adhering  to  pure  anal- 
ysis, of  which  Weierstrass,  Fuchs,  and  Frobenius  are  types.. 
The  work  of  Fuchs  and  the  theory  of  elementary  divisors  has- 
formed  the  basis  of  a  late  work  by  Sauvage  (1895).     Poincar^'s- 

*  Klein's  Evanston  Lectures,  Lect.  I. 

\  Klein's  Evanston  Lectures,  Lect.  II,  III.  ' 


Art.  12.J  infinite  series.  539 

recent  contributions  are  also  very  notable.  His  theory  of 
Fuchsian  equations  (also  investigated  by  Klein)  is  connected 
with  the  general  theory.  He  has  also  brought  the  whole  sub- 
ject into  close  relations  with  the  theory  of  functions.  Appell 
has  recently  contributed  to  the  theory  of  linear  differential 
equations  transformable  into  themselves  by  change  of  the  func- 
tion and  the  variable.  Helge  von  Koch  has  written  on  infinite 
determinants  and  linear  differential  equations.  Picard  has  un- 
dertaken the  generalization  of  the  work  of  Fuchs  and  Poincare 
in  the  case  of  differential  equations  of  the  second  order.  Fabry 
(1885)  has  generalized  the  normal  integrals  of  Thome,  integrals 
which  Poincar^  has  called  ''integrals  anormales,"  and  which 
Picard  has  recently  studied.  Riquier  has  treated  the  question 
of  the  existence  of  integrals  in  any  differential  system  and 
given  a  brief  summary  of  the  history  to  1895.*  The  number  of 
contributors  in  recent  times  is  very  great,  and  includes,  besides 
those  already  mentioned,  the  names  of  Brioschi,  Konigsberger, 
Peano,  Graf,  Hamburger,  Graindorge,  Schlafli,  Glaisher,  Lom- 
mel,  Gilbert,  Fabry,  Craig,  and  Autonne. 

Art.  12.    Infinite  Series. 

The  Theory  of  Infinite  Series  f  in  its  historical  develop- 
ment has  been  divided  by  Reiff  into  three  periods:  (i)  the 
period  of  Newton  and  Leibniz,  that  of  its  introduction ; 
(2)  that  of  Euler,  the  formal  period  ;  (3)  the  modern,  that  of 
the  scientific  investigation  of  the  validity  of  infinite  series,  a 
period  beginning  with  Gauss.  This  critical  period  begins  with 
the  publication  of  Gauss's  celebrated  memoir  on   the   series 

,+fL^^  +  ---(--+')-f-(^+'V  +  ....in    .812.     Euler 

'     I  .  K  I  .  2.  ;K.(r  H-  0 

*  Riquier,  C,  M6moire  sur  I'existence  des  int6grales  dans  un  syst^me  dif- 
f^rentiel  quelconque,  etc.     M^moires  des  Savants  6tran.e:ers,  Vol.  XXXII,  No.  3. 

f  Cantor,  M.,  Geschichte  der  Mathematik,  Vol.  Ill,  pp.  53.  7^  :  Reiff,  R., 
Gescbichte  der  unendlichen  Reihen,  TiibinRen,  1889  ;  Cajori,  F.,  Bulletin 
New  York  Mathematical  Society,  Vol.  I,  p.  184;  History  of  Teaching  of  Mathe- 
matics in  United  States,  p.  361. 


540  HISTORY    OF    MODERN    MATHEMATICS.  [ChaP.  XI. 

had  already  considered  this  series,  but  Gauss  was  the  first  to 
master  it,  and  under  the  name  *' hypergeometric  series"  (due 
to  Pfaff)  it  has  since  occupied  the  attention  of  Jacobi,  Kummer, 
Schwarz,  Cayley,  Goursat,  and  numerous  others.  The  partic- 
ular series  is  not  so  important  as  is  the  standard  of  criticism 
Avhich  Gauss  set  up,  embodying  the  simpler  criteria  of  con- 
vergence and  the  questions  of  remainders  and  the  range  of 
convergence. 

Gauss's  contributions  were  not  at  once  appreciated,  and 
the  next  to  call  attention  to  the  subject  was  Cauchy  (1821), 
who  may  be  considered  the  fotnder  of  the  theory  of  con- 
vergence and  divergence  of  series.  He  was  one  of  the  first  to 
insist  on  strict  tests  of  convergence ;  he  showed  that  if  two 
series  are  convergent  their  product  is  not  necessarily  so;  and 
with  him  begins  the  discovery  of  effective  criteria  of  converg- 
ence and  divergence.  It  should  be  mentioned,  however,  that 
these  terms  had  been  introduced  long  before  by  Gregory  (1668), 
that  Euler  and  Gauss  had  given  various  criteria,  and  that 
Maclaurin  had  anticipated  a  few  of  Cauchy's  discoveries. 
Cauchy  advanced  the  theory  of  power  series  by  his  expansion 
of  a  complex  function  in  such  a  form.  •  His  test  for  convergence 
is  still  one  of  the  most  satisfactory  when  the  integration  in- 
volved is  possible. 

Abel  was  the  next  important  contributor.     In  his  memoir 

ffl  fJtifH  I  ^ 

(1826)  on  the  series  i  +  —^  -\ ^^ — j -^^  -f"  •  •  •  ^^  corrected 

certain  of  Cauchy's  conclusions,  and  gave  a  completely  scien- 
tific summation  of  the  series  for  complex  values  of  m  and  x. 
He  was  emphatic  against  the  reckless  use  of  series,  and  showed 
the  necessity  of  considering  the  subject  of  continuity  in  ques- 
tions of  convergence. 

Cauchy's  methods  led  to  special  rather  than  general  criteria, 
and  the  same  may  be  said  of  Raabe  (1832),  who  made  the  first 
elaborate  investigation  of  the  subject,  of  De  Morgan  (from 
1842),  whose  logarithmic  test  DuBois-Reymond  (1873)  and 
Pringsheim  (1889)  have  shown  to  fail  within  a  certain  region ; 


Art.  12.J  INFINITE    SERIES.  541 

of  Bertrand  (1842),  Bonnet  (1843),  Malmsten  (1846,  1847,  the 
latter  without  integration);  Stokes  (1847),  Paucker  (1852). 
Tchebichef  (1852),  and  Arndt  1853).  General  criteria  began 
with  Kummer  (1835),  and  have  been  studied  by  Eisenstein 
(1847),  Weierstrass  in  his  various  contributions  to  the  theory 
of  functions,  Dini  (1867),  DuBois-Reymond  (1873),  and  many 
others.  Pringsheim's  (from  1889)  memoirs  present  the  most 
complete  general  theory. 

The  Theory  of  Uniform  Convergence  was  treated  by 
Cauchy  (1821),  his  limitations  being  pointed  out  by  Abel,  but 
the  first  to  attack  it  successfully  were  Stokes  and  Seidel 
(1847-48).  Cauchy  took  up  the  problem  again  (1853),  acknowl- 
edging Abel's  criticism,  and  reaching  the  same  conclusions 
which  Stokes  had  already  found.  Thome  used  the  doctrine 
(1866),  but  there  was  great  delay  in  recognizing  the  importance 
of  distinguishing  between  uniform  and  non-uniform  converg- 
ence, in  spite  of  the  demands  of  the  theory  of  functions. 

Semi-Convergent  Series  were  studied  by  Poisson  (1823), 
who  also  gave  a  general  form  for  the  remainder  of  the  Mac- 
laurin  formula.  The  most  important  solution  of  the  problem 
is  due,  however,  to  Jacobi  (1834),  who  attacked  the  question  of 
the  remainder  from  a  different  standpoint  and  reached  a  differ- 
ent formula.  This  expression  was  also  worked  out,  and 
another  one  given,  by  Malmsten  (1847).  Schlomilch  (Zeitschrift, 
Vol. I, p.  192, 1856)  also  improved  Jacobi's  remainder, and  showed 
the  relation  between  the  remainder  and  Bernoulli's  function 
F{x)  =  i«  +  2"  +  . .  .  +  (;ir  —  I)^  Genocchi  (1852)  has  further 
contributed  to  the  theory. 

Among  the  early  writers  was  Wronski,  whose  *'  loi  supreme  " 
(181 5)  was  hardly  recognized  until  Cayley  (1873)  brought  it 
into  prominence.  Transon  (1874),  Ch.  Lagrange  (1884),. 
Echols,  and  Dickstein*  have  published  of  late  various  memoirs 
on  the  subject. 

Interpolation  Formulas  have  been  given  by  various  writers 

*  Bibliotheca  Mathematica,  1892-94;  historical. 


542  HISTORY    OF    MODERN    MATHEMATICS.  [ChAP,  XL 

from  Newton  to  the  present  time.  Lagrange's  theorem  is  well 
known,  although  Euler  had  already  given  an  analogous  form, 
as  are  also  Olivier's  formula  (1827),  and  those  of  Minding 
(1830),  Cauchy  (1837),  Jacobi  (1845),  Grunert  (1850,  1853), 
Christoffel  (1858),  and  Mehler  (1864). 

Fourier's  Series*  were  being  investigated  as  the  result  of 
physical  considerations  at  the  same  time  that  Gauss,  Abel, 
and  Cauchy  were  working  out  the  theory  of  infinite  series. 
Series  for  the  expansion  of  sines  and  cosines,  of  multiple  arcs 
in  powers  of  the  sine  and  cosine  of  the  arc  had  been  treated 
by  Jakob  Bernoulli  (1702)  and  his  brother  Johann  (1701)  and 
still  earlier  by  Viete.  Euler  and  Lagrange  had  simplified  the 
subject,  as  have,  more  recently,  Poinsot,  Schroter,  Glaisher, 
and  Kummer.  Fourier  (1807)  set  for  himself  a  different  prob- 
lem, to  expand  a  given  function  of  x  in  terms  of  the  sines  or 
cosines  of  multiples  of  x,  a  problem  which  he  embodied  in  his 
Theorie  analytique  de  la  Chaleur  (1822).  Euler  had  already 
given  the  formulas  for  determining  the  coefficients  in  the 
series;  and  Lagrange  had  passed  over  them  without  recog- 
nizing their  value,  but  Fourier  was  the  first  to  assert  and  at- 
tempt to  prove  the  general  theorem.  Poisson  (1820-23)  also 
attacked  the  problem  from  a  different  standpoint.  Fourier 
did  not,  however,  settle  the  question  of  convergence  of  his 
series,  a  matter  left  for  Cauchy  (1826)  to  attempt  and  for 
Dirichlet  (1829)  to  handle  in  a  thoroughly  scientific  manner. 
Dirichlet's  treatment  (Crelle,  1829),  while  bringing  the  theory 
of  trigonometric  series  to  a  temporary  conclusion,  has  been 
the  subject  of  criticism  and  improvement  by  Riemann  (1854), 
Heine,  Lipschitz,  Schlafli,  and  DuBois-Reymond.  Among 
other  prominent  contributors  to  the  theory  of  trigonometric 
and  Fourier  series  have  been  Dini,  Hermite,  Halphen,  Krause, 
Byerly  and  Appell. 

*  Historical  Summary  by  Bocher,  Chap.   IX  of  Byerly's  Fourier's  Series 
and  Spherical    Harmonics,  Boston,  1893;  Sachse,    A.,  Essai  historique  sur  la 

representation    d'une    fonction par   une    serie    trigonom6trique.      Bulletin 

des  Sciences  mathematiques,  Part  I,  1880,  pp.  43,  83. 


Art.  13.1  THEORY    OF    FUNCTIONS.  543 

Art.  13.    Theory  of  Functions. 

The  Theory  of  Functions  *  may  be  said  to  have  its  first 
development  in  Newton's  works,  although  algebraists  had  al- 
ready become  familiar  with  irrational  functions  in  considering 
cubic  and  quartic  equations.  Newton  seems  first  to  have 
grasped  the  idea  of  such  expressions  in  his  consideration  of 
symmetric  functions  of  the  roots  of  an  equation.  The  word 
was  employed  by  Leibniz  (1694),  but  in  connection  with  the 
Cartesian  geometry.  In  its  modern  sense  it  seems  to  have 
been  first  used  by  Johann  Bernoulli,  who  distinguished  between 
algebraic  and  transcendent  functions.  He  also  used  (1718)  the 
function  symbol  0.  Clairaut  (1734)  used  IIx,  ^x,  Ax,  for  va- 
rious functions  of  x,  a  symbolism  substantially  followed  by 
d'Alembert  (1747)  and  Euler  (1753).  Lagrange  (1772,  1797, 
1806)  laid  the  foundations  for  the  general  theory,  giving  to 
the  symbol  a  broader  meaning,  and  to  the  symbols  /,  0,  i^,  .  .  . , 
f\  0',  F\  ,  .  .  their  modern  signification.  Gauss  contributed 
to  the  theory,  especially  in  his  proofs  of  the  fundamental 
theorem  of  algebra,  and  discussed  and  gave  name  to  the  theory 
of  "conforme  Abbildung,"  the  "  orthomorphosis  "  of  Cayley. 

Making  Lagrange's  work  a  point  of  departure,  Cauchy  so 
greatly  developed  the  theory  that  he  is  justly  considered  one 
of  its  founders.  His  memoirs  extend  over  the  period  1814- 
185 1,  and  cover  subjects  like  those  of  integrals  with  imaginary 
limits,  infinite  series  and  questions  of  convergence,  the  appHca- 
tion   of  the  infinitesimal   calculus  to  the  theory  of  complex 

*  Brill,  A.,  and  Noether,  M.,  Die  Entwickelung  der  Theorie  der  algebrai- 
schen  Functionen  in  alterer  und  neuerer  Zeit,  Bericht  erstattet  der  Deutschen 
Mathematiker-Vereinigung,  Jahresbericht,  Vol.  II, 'pp.  107-566,  Berlin,  1894; 
KOnigsberger,  L.»  Zur  Geschichte  der  Theorie  der  elliptischen  Transcendenten 
in  den  Jahren  1826-29,  Leipzig,  1879;  Williamson.  B  ,  Infinitesimal  Calculus, 
Encyclopaedia  Britannica;  Schlesinger,  L.,  Dififerentialgleichungen,  Vol.  I,  1895; 
Casorati,  F. ,  Teorica  delle  funzioni  di  variabili  complesse,  Vol.  I,  1868;  Klein's 
Evanston  Lectures.  For  bibliography  and  historical  notes,  see  Harkness  and 
Morley's  Theory  of  Functions,  1893,  and  Forsyth's  Theory  of  Functions,  1893: 
Enestrom.  G.,  Note  historique  sur  les  symboles.  .  .  .  Bibliotheca  Mathematica, 
1891,  p   89. 


544  HISTORY    OF    MODERN    MATHEMATICS.  [ChAP.  XL 

numbers,  the  investigation  of  the  fundamental  laws  of  mathe- 
matics, and  numerous  other  lines  which  appear  in  the  general 
theory  of  functions  as  considered  to-day.  Originally  opposed 
to  the  movement  started  by  Gauss,  the  free  use  of  complex 
numbers,  he  finally  became,  like  Abel,  its  advocate.  To  him 
is  largely  due  the  present  orientation  of  mathematical  research, 
making  prominent  the  theory  of  functions,  distinguishing  be- 
tween classes  of  functions,  and  placing  the  whole  subject  upon 
a  rigid  foundation.  The  historical  development  of  the  gen- 
eral theory  now  becomes  so  interwoven  with  that  of  special 
classes  of  functions,  and  notably  the  elliptic  and  Abelian,  that 
economy  of  space  requires  their  treatment  together,  and  hence 
a  digression  at  this  point. 

The  Theory  of  Elliptic  Functions*  is  usually  referred  for  its 
origin  to  Landen's  (1775)  substitution  of  two  elliptic  arcs  for  a 
single  hyperbolic  arc.  But  Jakob  Bernoulli  (1691)  had  sug- 
gested the  idea  of  comparing  non-congruent  arcs  of  the  same 
curve,  and  Johann  had  followed  up  the  investigation.  Fagano 
(1716)  had  made  similar  studies,  and  both  Maclaurin  (1742)  and 
d'Alembert  (1746)  had  come  upon  the  borderland  of  elliptic 
functions.  Euler  (from  1761)  had  summarized  and  extended 
the  rudimentary  theory,  showing  the  necessity  for  a  convenient 
notation  for  elliptic  arcs,  and  prophesying  (1766)  that  "such 
signs  will  afford  a  new  sort  of  calculus  of  which  I  have  here  at- 
tempted the  exposition  of  the  first  elements."  Euler's  inves- 
tigations continued  until  about  the  time  of  his  death  (1783), 
and  to  him  Legendre  attributes  the  foundation  of  the  theory. 
Euler  was  probably  never  aware  of  Landen's  discovery. 

It  is  to  Legendre,  however,  that  the  theory  of  elliptic  func- 
tions is  largely  due,  and  on  it  his  fame  to  a  considerable  degree 
depends.  His  earlier  treatment  (1786)  almost  entirely  sub- 
stitutes a  strict  analytic  for  the  geometric  method.  For  forty 
years  he  had  the  theory  in  hand,  his  labor  culminating  in  his 

*  Enneper,  A.,    Elliptische    Funktionen,  Theorie    und    Geschichte,    Halle, 
1890;    Konigsberger,  L.,  Zur  Geschichte  der  Theorie    der   elliptischen    Tran- 

sceiidenten  in  den  Jahren  1826-29,  Leipzig,  1879. 


Art.  13.]  THEORY    OF    FUNCTIONS.  545 

Traits  des  Fonctions  elliptiques  et  des  Int^grales  Euleriennes 
(1825-28).  A  surprise  now  awaiting  him  is  best  told  in  his 
own  words:  *' Hardly  had  my  work  seen  the  light — its  name 
could  scarcely  have  become  known  to  scientific  foreigners, — 
when  I  learned  with  equal  surprise  and  satisfaction  that  two 
young  mathematicians,  MM.  Jacobi  of  Konigsberg  and  Abel 
of  Christiania,  had  succeeded  by  their  own  studies  in  perfect- 
ing considerably  the  theory  of  elliptic  functions  in  its  highest 
parts."  Abel  began  his  contributions  to  the  theory  in  1825, 
and  even  then  was  in  possession  of  his  fundamential  theorem 
which  he  communicated  to  the  Paris  Academy  in  1826.  This 
communication  being  so  poorly  transcribed  was  not  published 
in  full  until  1841,  although  the  theorem  was  sent  to  Crelle 
(1829)  just  before  Abel's  early  death.  Abel  discovered  the 
double  periodicity  of  elliptic  functions,  and  with  him  began 
the  treatment  of  the  elliptic  integral  as  a  function  of  the 
amplitude. 

Jacobi,  as  also  Legendre  and  Gauss,  was  especially  cordial  in 
praise  of  the  delayed  theorem  of  the  youthful  Abel.  He  calls 
it  a  "monumentum  aere  perennius,"  and  his  name  ''das 
Abel'sche  Theorem  "  has  since  attached  to  it.  The  functions 
of  multiple  periodicity  to  which  it  refers  have  been  called 
Abelian  Functions.  Abel's  work  was  early  proved  and  eluci- 
dated by  Liouville  and  Hermite.  Serret  and  Chasles  in  the 
Comptes  Rendus,  Weierstrass  (1853),  Clebsch  and  Gordan  in 
their  Theorie  der  Abel'sche  Functionen  (1866),  and  Briot  and 
Bouquet  in  their  two  treatises  have  greatly  elaborated  the 
theory.  Riemann's  *  (1857)  celebrated  memoir  in  Crelle  pre- 
sented the  subject  in  such  a  novel  form  that  his  treatment  was 
slow  of  acceptance.  He  based  the  theory  of  Abelian  integrals 
and  their  inverse,  the  Abelian  functions,  on  the  idea  of  the  sur- 
face now  so  well  known  by  his  name,  and  on  the  correspond- 
ing fundamental  existence  theorems.     Clebsch,  starting  from 

*  Klein,  Evanston  Lectures,  p.  3  ;  Riemann  and  Modern  Mathematics, 
translated  by  Ziwet,  Bulletin  of  American  Mathematical  Society,  Vol.  I,  p.  165; 
Burkhardt,  H.,  Vortrag  iiber  Riemann,  Gottingen,  1892. 


546  HISTORY    OF    iMODERN    MATHEMATICS.  [Chap.  XI. 

an  algebraic  curve  defined  by  its  equation,  made  the  subject 
more  accessible,  and  generalized  the  theory  of  Abelian  integrals 
to  a  theory  of  algebraic  functions  with  several  variables,  thus 
creating  a  branch  which  has  been  developed  by  Noether, 
Picard,  and  Poincare.  The  introduction  of  the  theory  of  in- 
variants and  projective  geometry  into  the  domain  of  hyper- 
elliptic  and  Abelian  functions  is  an  extension  of  Clebsch's 
scheme.  In  this  extension,  as  in  the  general  theory  of  Abelian 
functions,  Klein  has  been  a  leader.  With  the  development  of 
the  theory  of  Abelian  functions  is  connected  a  long  list  of 
names,  including  those  of  Schottky,  Humbert,  C.  Neumann, 
Fricke,  Konigsberger,  Prym,  Schwarz,  Painlev^,  Hurwitz, 
Brioschi,  Borchardt,  Cayley,  Forsyth,  and  Rosenhain,  besides 
others  already  mentioned. 

Returning  to  the  theory  of  elliptic  functions,  Jacobi  (1827) 
began  by  adding  greatly  to  Legendre's  work.  He  created  a 
new  notation  and  gave  name  to  the  "  modular  equations  "  of 
which  he  made  use.  Among  those  who  have  written  treatises 
upon  the  elliptic-function  theory  are  Briot  and  Bouquet, 
Laurent,  Halphen,  Konigsberger,  Hermite,  Dur^ge,  and  Cayley. 
The  introduction  of  the  subject  into  the  Cambridge  Tripos 
(1873),  and  the  fact  that  Cayley 's  only  book  was  devoted  to  it, 
have  tended  to  popularize  the  theory  in  England. 

The  Theory  of  Theta  Functions  was  the  simultaneous  and 
independent  creation  of  Jacobi  and  Abel  (1828).  Gauss's 
notes  show  that  he  was  aware  of  the  properties  of  the  theta 
functions  twenty  years  earlier,  but  he  never  published  his  in- 
vestigations. Among  the  leading  contributors  to  the  theory, 
are  Rosenhain  (1846,  published  in  185 1)  and  Gopel  U847)>  who 
connected  the  double  theta  functions  with  the  theory  of  Abelian 
functions  of  two  variables  and  established  the  theory  of  hyper- 
elliptic  functions  in  a  manner  corresponding  to  the  Jacobian 
theory  of  eUiptic  functions.  Weierstrass  has  also  developed 
the  theory  of  theta  functions  independently  of  the  form  of  their 
space  boundaries,  researches  elaborated  by  Konigsberger  (1865) 
to  give  the  addition   theorem.     Riemann  has  completed  the 


Art.  13.]  THEORY    OF    FUNCTIONS.  54? 

investigation  of  the  relation  between  the  theory  of  the  theta 
and  the  Abelian  functions,  and  has  raised  theta  functions  to 
their  present  position  by  making  them  an  essential  part  of  his 
theory  of  Abelian  integrals.  H.  J.  S.  Smith  has  included 
among  his  contributions  to  this  subject  the  theory  of  omega 
functions.  Among  the  recent  contributors  are  Krazer  and 
Prym  (1892),  and  Wirtinger  (1895). 

Cayley  was  a  prominent  contributor  to  the  theory  of 
periodic  functions.  His  memoir  (1845)  o"  doubly  periodic 
functions  extended  Abel's  investigations  on  doubly  infinite 
products.  Euler  had  given  singly  infinite  products  for  sin  x, 
cos  X,  and  Abel  had  generalized  these,  obtaining  for  the 
elementary  doubly  periodic  functions  expressions  for  sn  ;r, 
en  X,  dn  x.  Starting  from  these  expressions  of  Abel's  Cayley 
laid  a  complete  foundation  for  his  theory  of  elliptic  functions. 
Eisenstein  (1847)  followed,  giving  a  discussion  from  the  stand- 
point of  pure  analysis,  of  a  general  doubly  infinite  product, 
and  his  labors,  as  supplemented  by  Weierstrass,  are  classic. 

The  General  Theory  of  Functions  has  received  its  present 
form  largely  from  the  works  of  Cauchy,  Riemann,  and  Weier- 
strass. Endeavoring  to  subject  all  natural  laws  to  interpreta- 
tion by  mathematical  formulas,  Riemann  borrowed  his  methods 
from  the  theory  of  the  potential,  and  found  his  inspiration  in 
the  contemplation  of  mathematics  from  the  standpoint  of  the 
concrete.  Weierstrass,  on  the  other  hand,  proceeded  from  the 
purely  analytic  point  of  view.  To  Riemann*  is  due  the  idea 
of  making  certain  partial  differential  equations,  which  express 
the  fundamental  properties  of  all  functions,  the  foundation  of 
a  general  analytical  theory,  and  of  seeking  criteria  for  the 
determination  of  an  analytic  function  by  its  discontinuities 
and  boundary  conditions.  His  theory  has  been  elaborated  by 
Klein  (1882,  and  frequent  memoirs)  who  has  materially  ex- 
tended the  theory  of  Riemann's  surfaces.  Clebsch,  Luroth, 
and  later  writers  have  based  on  this  theory  their  researches  on 

*  Klein,    F..    Riemann  and    Modern    Mathematics,    translated    by    Ziwet, 
Bulletin  of  American  Mathematical  Society,  Vol.  I,  p.  165. 


548  HISTORY    OF    MODERN    xMATHEM ATICS.  [Chap.  XL 

loops.  Riemann's  speculations  were  not  without  weak  points^ 
and  these  have  been  fortified  in  connection  with  the  theory  of 
the  potential  by  C.  Neumann,  and  from  the  analytic  standpoint 
by  Schwarz. 

In  both  the  theory  of  general  and  of  elliptic  and  other 
functions,  Clebsch  was  prominent.  He  introduced  the  system- 
atic consideration  of  algebraic  curves  of  deficiency  I,  bring- 
ing to  bear  on  the  theory  of  eUiptic  functions  the  ideas  of 
modern  projective  geometry.  This  theory  Klein  has  gener- 
alized in  his  Theorie  der  elliptischen  Modulfunctionen,  and 
has  extended  the  method  to  the  theory  of  hypereUiptic  and 
Abelian  functions. 

Following  Riemann  came  the  equally  fundamental  and 
original  and  more  rigorously  worked  out  theory  of  Weierstrass. 
His  early  lectures  on  functions  are  justly  considered  a  land- 
mark in  modern  mathematical  development.  In  particular, 
his  researches  on  Abelian  transcendents  are  perhaps  the  most 
important  since  those  of  Abel  and  Jacobi.  His  contributions 
to  the  theory  of  elHptic  functions,  including  the  introduction 
of  the  function  l>{u),  are  also  of  great  importance.  His  con- 
tributions to  the  general  function  theory  include  much  of  the 
symbolism  and  nomenclature,  and  many  theorems.  He  first 
announced  (1866)  the  existence  of  natural  limits  for  analytic 
functions,  a  subject  further  investigated  by  Schwarz,  Klein^ 
and  Fricke.  He  developed  the  theory  of  functions  of  complex 
variables  from  its  foundations,  and  his  contributions  to  the 
theory  of  functions  of  real  variables  were  no  less  marked. 

Fuchs  has  been  a  prominent  contributor,  in  particular  (1872) 
on  the  general  form  of  a  function  with  essential  singularities. 
On  functions  with  an  infinite  number  of  essential  singularities 
Mittag-Leffler  (from  1882)  has  written  and  contributed  a 
fundamental  theorem.  On  the  classification  of  singularities  of 
functions  Guichard  (1883)  has  summarized  and  extended  the 
researches,  and  Mittag-Lefifler  and  G.  Cantor  have  contributed 
to  the  same  result.  Laguerre  (from  1882)  was  the  first  to 
discuss  the  *'  class "   of  transcendent   functions,  a  subject  to 


Art.  13.J  THEORY  of  functions.  549 

which  Poincar^,  Cesaro,  Vivanti,  and  Hermite  have  also  con- 
tributed. Automorphic  functions,  as  named  by  Klein,  have 
been  investigated  chiefly  by  Poincare,  who  has  established  their 
general  classification.  The  contributors  to  the  theory  include 
Schwarz,  Fuchs,  Cayley,  Weber,  Schlesinger,  and  Burnside. 

The  Theory  of  Elliptic  Modular  Functions,  proceding  from 
Eisenstein's  memoir  (1847)  ^^^  the  lectures  of  Weierstrass  on 
elliptic  functions,  has  of  late  assumed  prominence  through  the 
influence  of  the  Klein  school.  Schlafli  (1870),  and  later  Klein, 
Dyck,  Gierster,  and  Hurwitz,  have  worked  out  the  theory 
which  Klein  and  Fricke  have  embodied  in  the  recent  Vorle- 
sungen  iiber  die  Theorie  der  elliptischen  Modulfunctionen 
(1890-92).  In  this  theory  the  memoirs  of  Dedekind  (1877), 
Klein  (1878),  and  Poincar^  (from  1881)  have  been  among  the 
most  prominent. 

For  the'names  of  the  leading  contributors  to  the  generaj 
and  special  theories,  including  among  others  Jordan,  Hermite, 
Holder,  Picard,  Biermann,  Darboux,  Pellet,  Reichardt,  Burk- 
hardt,  Krause,  and  Humbert,  reference  must  be  had  to  the 
Brill-Noether  Bericht. 

Of  the  various  special  algebraic  functions  space  allows  men- 
tion of  but  one  class,that  bearing  Bessel's  name.  Bessel's  func- 
tions *  of  the  zero  order  are  found  in  memoirs  of  Daniel  Ber- 
noulH  (1732)  and  Euler  (1764),  and  before  the  end  of  the  eigh- 
teenth century  all  the  Bessel  functions  of  the  first  kind  and 
integral  order  had  been  used.  Their  prominence  as  special 
functions  is  due,  however,  to  Bessel  (1816-17),  who  put  them 
in  their  present  form  in  1824.  Lagrange's  series  (1770),  with 
Laplace's  extension  (1777),  had  been  regarded  as  the  best 
method  of  solving  Kepler's  problem  (to  express  the  variable 
quantities  in  undisturbed  planetary  motion  in  terms  of  the 
time  or  mean  anomaly),  and  to  improve  this  method  Bessel's 
functions  were  first  prominently  used.  Hankel  (1869),  Lom- 
mel  (from  1868),  F.  Neumann,  Heine,  Graf  (1893),  Gray  and 

*  Bocher,  M.,  A  bit  of  mathematical  history,  Bulletin  of  New  York  Mathe- 
matical Society,  Vol.  II,  p.  107. 


550  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XL 

Mathews  (1895),  and  others  have  contributed  to  the  theory. 
Lord  Rayleigh  (1878)  has  shown  the  relation  between  Bessel's 
and  Laplace's  functions,  but  they  are  nevertheless  looked  upon 
as  a  distinct  system  of  transcendents.  Tables  of  Bessel's  func- 
tions were  prepared  by  Bessel  (1824),  by  Hansen  (1843),  and 
by  Meissel  (1888). 

Art.  14.    Probabilities  and  Least  Squares. 

The  Theory  of  Probabilities  and  Errors^  is,  as  applied  to 
observations,  largely  a  nineteenth-century  development.  The 
doctrine  of  probabilities  dates,  however,  as  far  back  as  Fermat 
and  Pascal  (1654).  Huygens  (1657)  gave  the  first  scientific 
treatment  of  the  subject,  and  Jakob  Bernoulli's  Ars  Conjectandi 
(posthumous,  171 3)  and  De  Moivre's  Doctrine  of  Chances 
(171 8)traised  the  subject  to  the  plane  of  a  branch  of  mathematics. 
The  theory  of  errors  may  be  traced  back  to  Cotes's  Opera 
Miscellanea  (posthumous,  1722),  but  a  memoir  prepared  by 
Simpson  in  1755  (printed  1756)  first  applied  the  theory  to  the 
discussion  of  errors  of  observation.  The  reprint  (1757)  of  this 
memoir  lays  down  the  axioms  that  positive  and  negative  errors 
are  equally  probable,  and  that  there  are  certain  assignable 
limits  within  which  all  errors  may  be  supposed  to  fall ;  con- 
tinuous errors  are  discussed  and  a  probability  curve  is  given. 
Laplace  (1774)  made  the  first  attempt  to  deduce  a  rule  for  the 
combination  of  observations  from  the  principles  of  the  theory 
of  probabilities.  He  represented  the  law  of  probability  of 
errors  by  a  curve  y  =  (p{x),  x  being  any  error  and  y  its  proba- 
bility, and  laid  down  three  properties  of  this  curve:  (i)  It  is 
symmetric  as  to  the  jj/-axis;  (2)  the  ;r-axis  is  an  asymptote,  the 
probability  of  the  error  00  being  o;  (3)  the  area  enclosed  is  i, 
it  being  certain  that  an  error  exists.     He  deduced  a  formula 

*  Merriman,  M.,  Method  of  Least  Squares,  New  York,  1884,  p.  182  ;  Trans- 
actions of  Connecticut  Academy,  1877,  Vol.  IV,  p.  151,  with  complete  bibliog- 
raphy; Todhunter,  I.,  History  of  the  Mathematical  Theory  of  Probability,. 
1865;  Cantor,  M.,  Geschichte  der  Mathematik,  Vol.  Ill,  p.  316. 

J-EnestrOm,  G.,  Review  of  Cantor,  Bibliotheca  Mathematica,  1896,  p.  20. 


Art.  14.J  PROBABILITIES    AND    LEAST    SQUARES.  551 

for  the  mean  of  three  observations.  He  also  gave  (1781)  a 
formula  for  the  law  of  facility  of  error  (a  term  due  to  Lagrange, 
1774),  but  one  which  led  to  unmanageable  equations.  Daniel 
Bernoulli  (1778)  introduced  the  principle  of  the  maximum 
product  of  the  probabilities  of  a  system  of  concurrent  errors. 

The  Method  of  Least  Squares  is  due  to  Legendre  (1805), 
who  introduced  it  in  his  Nouvelles  methodes  pour  la  determi- 
nation des  orbites  des  cometes.  In  ignorance  of  Legendre's 
contribution,  an  Irish-American  writer,  Adrain,  editoT  of  "  The 
Analyst"  (1808),  first  deduced  the  law  of  facility  of  error, 
0(4r)  =  6-^~*'^*',  c  and  h  being  constants  depending  on  pre- 
cision of  observation.  He  gave  two  proofs,  the  second  being 
essentially  the  same  as  Herschcl's  (1850).  Gauss  gave  the  first 
proof  which  seems  to  have  been  known  in  Europe  (the  third 
after  Adrain's)  in  1809.  To  him  is  due  much  of  the  honor  of 
placing  tlie  subject  before  the  mathematical  world,  both  as  to 
the  theory  and  its  applications. 

Further  proofs  were  given  by  Laplace  (1810,  1812),  Gauss 
(1823),  Ivory  (1825,  1826),  Hagen  (1837),  Bessel  (1838),  Donkin 
(1844,  1856),  and  Crofton  (1870).  Other  contributors  have 
been  Ellis  (1844),  De  Morgan  (1864),  Glaisher  (1872),  and  Schi- 
aparelli  (1875).  Peter's  (1856)  formula  for  r,  the  probable  error 
of  a  single  observation,  is  well  known.* 

Among  the  contributors  to  the  general  theory  of  probabil- 
ities in  the  nineteenth  century  have  been  Laplace,  Lacroix 
(1816),  Littrow  (1833),  Quetelet  (1853),  Dedekind  (i860),  Hel- 
mert  (1872),  Laurent  (1873),  Liagre,  Didion,  and  Pearson. 
De  Morgan  and  Boole  improved  the  theory,  but  added  little 
that  was  fundamentally  new.  Czuber  has  done  much  both  in  his 
own  contributions  (1884,  1891),  and  in  his  translation  (1879) 
of  Meyer.  On  the  geometric  side  the  influence  of  Miller  and 
The  Educational  Times  has  been  marked,  as  also  that  of  such 
contributors  to  this  journal  as  Crofton,  McColl,  Wolstenholme, 
Watson,  and  Artemas  Martin. 

*  Bulletin  of  New  York  Mathematical  Society,  Vol.  11.  p.  57. 


552  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  Xi. 

Art.  15.    Analytic  Geometry. 

The  History  of  Geometry^  may  be  roughly  divided  into 
the  four  periods:  (i)  The  synthetic  geometry  of  the  Greeks, 
practically  closing  with  Archimedes  ;  (2)  The  birth  of  analytic 
geometry,  in  which  the  synthetic  geometry  of  Guldin,  De- 
sargues,  Kepler, and  Roberval  merged  into  the  coordinate  geom- 
etry of  Descartes  and  Fermat ;  (3)  1650  to  i8cx),  characterized 
by  the  application  of  the  calculus  to  geometry,  and  including 
the  names  of  Newton,  Leibnitz,  the  Bernoullis,  Clairaut,  Mac- 
laurin,  Euler,  and  Lagrange,  each  an  analyst  rather  than  a  ge- 
ometer; (4)  The  nineteenth  century,  the  renaissance  of  pure 
geometry,  characterized  by  the  descriptive  geometry  of  Monge, 
the  modern  synthetic  of  Poncelet,  Steiner,  von  Staudt,  and 
Cremona,  the  modern  analytic  founded  by  Plucker,  the  non- 
Euclidean  hypothesis  of  Lobachevsky  and  Bolyai,  and  the 
more  elementary  geometry  of  the  triangle  founded  by  Lemoine. 
It  is  quite  impossible  to  draw  the  line  between  the  analytic 
and  the  synthetic  geometry  of  the  nineteenth  century,  in  their 
historical  development,  and  Arts.  15  and  16  should  be  read  to- 
gether. 

The  Analytic  Geometry  which  Descartes  gave  to  the  world 
in  1637  was  confined  to  plane  curves,  and  the  various  important 
properties  common  to  all  algebraic  curves  were  soon  discovered. 
To  the  theory  Newton  contributed  three  celebrated  theorems 
on  the  Enumeratio  linearum  tertii  ordinis  f  (1706),  while  others 
are  due  to  Cotes  (1722),  Maclaurin,  and  Waring   (1762,  1772, 

*  Loria,  G.,  II  passato  e  il  presente  delle  principali  teorie  geometriche. 
Memorie  Accademia  Torino,  1887;  translated  into  German  by  F.  Schiitte 
under  the  title  Die  hauptsachlichsten  Theorien  der  Geometrie  in  ihrer  friiheren 
und  heutigen  Entwickelung,  Leipzig,  1888;  Chasles.  M.,  Aper9u  historique 
sur  Torigine  et  le  d6veloppement  des  m6thodes  en  Geometric,  1889  ;  Chasles, 
M.,  Rapport  sur  les  Progres  de  la  Geometric,  Paris,  1870  ;  Cayley,  A,,  Curves, 
Encyclopaedia  Britannica;  Klein,  F.,  Evanston  Lectures  on  Mathematics,  New 
York,  1894;  A.  V.  Braunmiihl,  Historische  Studie  iiber  die  organische  Er- 
zeugung  ebener  Curven,  Dyck's  Katalog  mathematischer  Modelle,  1892. 

f  Ball,  W.  W.  R.,  On  Newton's  classification  of  cubic  curves.  Transactions 
of  London  Mathematical  Society,  1891,  p.  104. 


Art.  15.1  analytic  geometry.  553 

etc.)-  The  scientific  foundations  of  the  theory  of  plane  curves 
may  be  ascribed,  however,  to  Euler  (1748)  and  Cramer  (1750). 
Euler  distinguished  between  algebraic  and  transcendent  curves, 
-and  attempted  a  classification  of  the  former.  Cramer  is  well 
known  for  the  "  paradox  "  which  bears  his  name,  an  obstacle 
which  Lame  (1818)  finally  removed  from  the  theory.  To 
Cramer  is  also  due  an  attempt  to  put  the  theory  of  singulari- 
ties of  algebraic  curves  on  a  scientific  foundation,  although  in 
a  modern  geometric  sense  the  theory  was  first  treated  by 
Poncelet. 

Meanwhile  the  study  of  surfaces  was  becoming  prominent. 
Descartes  had  suggested  that  his  geometry  could  be  extended 
to  three-dimensional  space,  Wren  (1669)  had  discovered  the 
two  systems  of  generating  lines  on  the  hyperboloid  of  one 
sheet,  and  Parent  (1700)  had  referred  a  surface  to  three  coor- 
dinate planes.  The  geometry  of  three  dimensions  began  to 
assume  definite  shape,  however,  in  a  memoir  of  Clairaut's  (1731), 
in  which,  at  the  age  of  sixteen,  he  solved  with  rare  elegance 
many  of  the  problems  relating  to  curves  of  double  curvature. 
Euler  (1760)  laid  the  foundations  for  the  analytic  theory  of 
■curvature  of  surfaces,  attempting  the  classification  of  those 
of  the  second  degree  as  the  ancients  had  classified  curves 
of  the  second  order.  Monge,  Hachette,  and  other  members  of 
that  school  entered  into  the  study  of  surfaces  with  great  zeal. 
Monge  introduced  the  notion  of  families  of  surfaces,  and  dis- 
•covered  the  relation  between  the  theory  of  surfaces  and  the 
integration  of  partial  differential  equations,  enabling  each  to  be 
advantageously  viewed  from  the  standpoint  of  the  other.  The 
theory  of  surfaces  has  attracted  a  long  list  of  contributors  in 
the  nineteenth  century,  including  most  of  the  geometers  whose 
names  are  mentioned  in  the  present  article."^ 

Mobius  began  his  contributions  to  geometry  in  1823,  and 
four  years  later  published  his  Barycentrische  Calcul.  In  this 
great  work  he  introduced  homogeneous  coordinates  with  the 

*For  details  see  Loria,  II  passato  e  il  presente,  etc. 


554  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XI. 

attendant  symmetry  of  geometric  formulas,  the  scientific 
exposition  of  the  principle  of  signs  in  geometry,  and  the 
establishment  of  the  principle  of  geometric  correspondence 
simple  and  multiple.  He  also  (1852)  summed  up  the  classifi- 
cation of  cubic  curves,  a  service  rendered  by  Zeuthen  (1874) 
for  quartics.  To  the  period  of  Mobius  also  belong  Bobillier 
(1827),  who  first  used  trilinear  coordinates,  and  Bellavitis,  whose 
contributions  to  analytic  geometry  were  extensive.  Ger- 
p-onne's  labors  are  mentioned  in  the  next  article. 

o 

Of  all  modern  contributors  to  analytic  geometry,  Pliicker 
stands  foremost.  In  1828  he  published  the  first  volume  of  his 
Analytisch-geometrische  Entwickelungen,  in  which  appeared 
the  modern  abridged  notation,  and  which  marks  the  beginning 
of  a  new  era  for  analytic  geometry.  In  the  second  volume 
(1831)  he  sets  forth  the  present  analytic  form  of  the  principle 
of  duality.  To  him  is  due  (1833)  the  general  treatment  of  foci 
for  curves  of  higher  degree,  and  the  complete  classification  of 
plane  cubic  curves  (1835)  whici*.  had  been  so  frequently  tried 
before  him.  He  also  gave  (1839)  ^^  enumeration  of  plane 
curves  of  the  fourth  order,  which  Bragelogne  and  Euler  had 
attempted.  In  1842  he  gave  his  celebrated '*  six  equations" 
by  which  he  showed  that  the  characteristics  of  a  curve  (order^ 
class,  number  of  double  points,  number  of  cusps,  number  of 
double  tangents,  and  number  of  inflections)  are  known 
when  any  three  are  given.  To  him  is  also  due  the  first  scien- 
tific dual  definition  of  a  curve,  a  system  of  tangential  coordi- 
nates, and  an  investigation  of  the  question  of  double  tangents, 
a  question  further  elaborated  by  Cayley  (1847,  1858),  Hesse 
(1847),  Salmon  (1858),  and  Dersch  (1874).  The  theory  of 
ruled  surfaces,  opened  by  Monge,  was  also  extended  by  him. 
Possibly  the  greatest  service  rendered  by  Plucker  was  the  in- 
troduction of  the  straight  line  as  a  space  element,  his  first 
contribution  (1865)  being  followed  by  his  well-known  treatise 
on  the  subject  (1868-69).  In  this  work  he  treats  certain  general, 
properties  of  complexes,  congruences,  and  ruled  surfaces,  as. 
well  as  special  properties  of  linear  complexes  and  congruen-^ 


Art.  lo.]  ANALYTIC    GEOMETRY.  555 

ces,  subjects  also  considered  by  Kummer  and  by  Klein  and 
others  of  the  modern  school.  It  is  not  a  little  due  to  Plucker 
that  the  concept  of  4-  and  hence  /^-dimensional  space,  already 
suggested  by  Lagrange  and  Gauss,  became  the  subject  of 
later  research.  Riemann,  Helmholtz,  Lipschitz,  Kronecker, 
Klein,  Lie,  Veronese,  Cayley,  d'Ovidio,  and  many  others  have 
elaborated  the  theory.  The  regular  hypersoHds  in  4-dimen- 
sional  space  have  been  the  subject  of  special  study  by  Scheffler,, 
Rudel,  Hoppe,  Schlegel,  and  Stringham. 

Among  Jacobi's  contributions  is  the  consideration  (1836)  of 
curves  and  groups  of  points  resulting  from  the  intersection  of 
algebraic  surfaces,  a  subject  carried  forward  by  Reye  (1869). 
To  Jacobi  is  also  due  the  conformal  representation  of  the 
ellipsoid  on  a  plane,  a  treatment  completed  by  Schering  (1858). 
The  number  of  examples  of  conformal  representation  of  sur- 
faces on  planes  or  on  spheres  has  been  increased  by  Schwarz 
(1869)  and  Amstein  (1872). 

In  1844  Hesse,  whose  contributions  to  geometry  in  general 
are  both  numerous  and  valuable,  gave  the  complete  theory  of 
inflections  of  a  curve,  and  introduced  the  so-called  Hessian 
curve  as  the  first  instance  of  a  covariant  of  a  ternary  form. 
He  also  contributed  to  the  theory  of  curves  of  the  third  order, 
and  generalized  the  Pascal  and  Brianchon  theorems  on  a 
spherical  surface.  Hesse's  methods  have  recently  been  elabo- 
rated by  Gundelfinger  (1894). 

Besides  contributing  extensively  to  synthetic  geometry,, 
Chasles  added  to  the  theory  of  curves  of  the  third  and  fourth 
degrees.  In  the  method  of  characteristics  which  he  worked 
out  maybe  found  the  first  trace  of  the  Abzahlende  Geometrie* 
which  has  been  developed  by  Jonquieres,  Halphen  (1875),  and 
Schubert  (1876,  1879),  ^"^  ^^  which  Clebsch,  Lindemann,  and 
Hurwitz  have  also  contributed.  The  general  theory  of  corre- 
spondence starts  with  Geometry,  and  Chasles  (1864)  undertook 

*  Loria,    G.,    Notizie   storiche   sulla   Geometria   numerativa.     Bibliotheca. 
Maihemaiica,  1888,  pp.  39,  67  ;  i88g,  p.  23. 


556  HISTORY    OF    MODERN    MATHEMATICS.  [ChAP.  XI. 

the  first  special  researches  on  the  correspondence  of  algebraic 
curves,  limiting  his  investigations,  however,  to  curves  of  defi- 
ciency zero.  Cayley  (1866)  carried  this  theory  to  curves  of 
higher  deficiency,  and  Brill  (from  1873)  completed  the  theory. 
Cayley's  *  influence  on  geometry  was  very  great.  He  early 
carried  on  Pliicker's  consideration  of  singularities  of  a  curve, 
and  showed  (1864,  1866)  that  every  singularity  may  be  con- 
sidered as  compounded  of  ordinary  singularities  so  that  the 
*'  six  equations  "  apply  to  a  curve  with  any  singularities  what- 
soever. He  thus  opened  a  field  for  the  later  investigations  of 
Noether,  Zeuthen,  Halphen,  and  H.  J.  S.  Smith.  Cayley's 
theorems  on  the  intersection  of  curves  (1843)  ^^^  the  deter- 
mination of  self-corresponding  points  for  algebraic  correspond- 
ences of  a  simple  kind  are  fundamental  in  the  present  theory, 
subjects  to  which  Bacharach,  Brill,  and  Noether  have  also  con- 
tributed extensively.  Cayley  added  much  to  the  theories  of 
rational  transformation  and  correspondence,showingthe  distinc- 
tion between  the  theory  of  transformation  of  spaces  and  that  of 
correspondence  of  loci.  His  investigations  on  the  bitangents  of 
plane  curves,  and  in  particular  on  the  twenty-eight  bitangents  of 
a  non-singular  quartic,  his  developments  of  Pliicker's  conception 
of  foci,  his  discussion  of  the  osculating  conies  of  curves  and  of 
the  sextactic  points  on  a  plane  curve,  the  geometric  theory 
of  the  invariants  and  covariants  of  plane  curves,  are  all  note- 
worthy. He  was  the  first  to  announce  (1849)  the  twenty-seven 
lines  which  lie  on  a  cubic  surface,  he  extended  Salmon's  theory  of 
reciprocal  surfaces,  and  treated  (1869)  the  classification  of  cubic 
surfaces,  a  subject  already  discussed  by  Schlafli.  He  also  con- 
tributed to  the  theory  of  scrolls  (skew-ruled  surfaces),  orthog- 
onal systems  of  surfaces,  the  wave  surface,  etc.,  and  was  the 
first  to  reach  (1845)  ^^ly  very  general  results  in  the  theory  of 
curves  of  double  curvature,  a  theory  in  which  the  next  great 
advance  was  made  (1882)  by  Halphen  and  Noether.  Among 
Cayley's  other  contributions  to  geometry  is  his  theory  of  the 
Absolute,  a  figure  in  connection  with  which  all  metrical  prop- 
erties of  a  figure  are  considered. 

*  Biographical  Notice  in  Cayley's  Collected  papers,  Vol.  VIII. 


Art.  15.J  analytic  geometry.  557 

Clebsch*  was  also  prominent  in  the  study  of  curves  and 
surfaces.  He  first  applied  the  algebra  of  linear  transformation  to 
geometry.  He  emphasized  the  idea  of  deficiency  (Geschlecht) 
of  a  curve,  a  notion  which  dates  back  to  Abel,  and  appHed  the 
theory  of  elliptic  and  Abelian  functions  to  geometry,  using  it 
for  the  study  of  curves.  Clebsch  (1872)  investigated  the  shapes 
of  surfaces  of  the  third  order.  Following  him,  Klein  attacked 
the  problem  of  determining  all  possible  forms  of  such  surfaces, 
and  established  the  fact  that  by  the  principle  of  continuity  all 
forms  of  real  surfaces  of  the  third  order  can  be  derived  from 
the  particular  surface  having  four  real  conical  points.  Zeuthen 
(1874)  has  discussed  the  various  forms  of  plane  curves  of  the 
fourth  order,  showing  the  relation  between  his  results  and 
those  of  Klein  on  cubic  surfaces.  Attempts  have  been  made 
to  extend  the  subject  to  curves  of  the  ;^th  order,  but  no  gen- 
eral classification  has  been  made.  Quartic  surfaces  have  been 
studied  by  Rohn  (1887)  t>ut  without  a  complete  enumeration, 
and  the  same  writer  has  contributed  (1881)  to  the  theory  of 
Kummer  surfaces. 

Lie  has  adopted  Plucker's  generalized  space  element  and  ex- 
tended the  theory.  His  sphere  geometry  treats  the  subject 
from  the  higher  standpoint  of  six  homogeneous  coordinates, 
as  distinguished  from  the  elementary  sphere  geometry  with 
but  five  and  characterized  by  the  conformal  group,  a  geometry 
studied  by  Darboux.  Lie's  theory  of  contact  transformations, 
with  its  application  to  differential  equations,  his  line  and 
sphere  complexes,  and  his  work  on  minimum  surfaces  are  all 
prominent. 

Of  great  help  in  the  study  of  curves  and  surfaces  and  of 
the  theory  of  functions  are  the  models  prepared  by  Dyck, 
Brill,  O.  Henrici,  Schwarz,  Klein,  Schonflies,  Kummer,  and 
others.f 

The  Theory  of  Minimum  Surfaces  has  been  developed  along 

*  Klein,  Evanston  Lectures,  Lect.  I. 

fDyck,  W.,  Katalog  mathematischer  und  mathematisch-physikalischer 
Modelle,  Miinchen,  1892  ;  Deutsche  Universitatsausstellung,  Mathematical 
Papers  of  Chicago  Congress,  p.  49. 


558  HISTORY    OF    MODERN    MATHEMATICS.  [ChaP.  XI. 

with  the  analytic  geometry  in  general.  Lagrange  (1760-61) 
gave  the  equation  of  the  minimum  surface  through  a  given 
contour,  and  Meusnier  (1776,  published  in  1785)  also  studied 
the  question.  But  from  this  time  on  for  half  a  century  little 
that  is  noteworthy  was  done,  save  by  Poisson  (18 13)  as  to  cer- 
tain imaginary  surfaces.  Monge  (1784)  and  Legendre  (1787) 
connected  the  study  of  surfaces  with  that  of  differential  equa- 
tions, but  this  did  not  immediately  affect  this  question.  Scherk 
(1835)  added  a  number  of  important  results,  and  first  applied 
the  labors  of  Monge  and  Legendre  to  the  theory.  Catalan 
(1842),  Bjorling  (1844),  and  Dini  (1865)  have  added  to  the 
subject.  But  the  most  prominent  contributors  have  been 
Bonnet,  Schwarz,  Darboux,  and  Weierstrass.  Bonnet  (from 
1853)  has  set  forth  a  new  system  of  formulas  relative  to  the 
general  theory  of  surfaces,  and  completely  solved  the  problem 
of  determining  the  minimum  surface  through  any  curve  and 
admitting  in  each  point  of  this  curve  a  given  tangent  plane. 
Weierstrass  (1866)  has  contributed  several  fundamental  theo- 
rems, has  shown  how  to  find  all  of  the  real  algebraic  minimum 
surfaces,  and  has  shown  the  connection  between  the  theory  of 
functions  of  an  imaginaay  variable  and  the  theory  of  minimum 
surfaces. 

Art.  16.    Modern  Geometry. 

Descriptive,*  Projective,  and  Modern  Synthetic  Geometry 
are  so  interwoven  in  their  historic  development  that  it  is  even 
more  difficult  to  separate  them  from  one  another  than  from 
the  analytic  geometry  just  mentioned.  Monge  had  been  in 
possession  of  his  theory  for  over  thirty  years  before  the  publi- 
cation of  his  Geom6trie  Descriptive  (1800),  a  delay  due  to  the 
jealous  desire  of  the  military  authorities  to  keep  the  valuable 
secret.  It  is  true  that  certain  of  its  features  can  be  traced 
back  to  Desargues,  Taylor,  Lambert,  and  Fr^zier,  but  it  was 
Monge    who  worked   it  out  in  detail    as  a  science,  although 

*  Wiener,  Chr. ,  Lehrbuch  der  darstellenden  Geometrie,  Leipzig,  1884-87; 
Geschichte  der  darstellenden  Geometrie,  1884. 


Art.  16.]  MODERN    GEOiMETRY.  '  559 

Lacroix  (1795),  inspired  by  Monge's  lectures  in  the  Ecole 
Polytechnique,  published  the  first  work  on  the  subject.  After 
Monge's  work  appeared,  Hachette  (1812,  1818,  1821)  added 
materially  to  its  symmetry,  subsequent  French  contributors 
being  Leroy  (1842),  Olivier  (from  1845),  ^^  la  Gournerie  (from 
i860),  Vallee,  de  Fourcy,  Adhemar,  and  others.  In  Germany 
leading  contributors  have  been  Ziegler  (1843),  Anger  (1858), 
and  especially  Fiedler  (3d  edn.  1883-88)  and  Wiener  (1884-87). 
At  this  period  Monge  by  no  means  confined  himself  to  the  de- 
scriptive geometry.  So  marked  were  his  labors  in  the  analytic 
geometry  that  he  has  been  called  the  father  of  the  modern 
theory.  He  also  set  forth  the  fundamental  theorem  of  recip- 
rocal polars,  though  not  in  modern  language,  gave  some  treat- 
ment of  ruled  surfaces,  and  extended  the  theory  of  polars  to 
quadrics.* 

Monge  and  his  school  concerned  themselves  especially  with 
the  relations  of  form,  and  particularly  with  those  of  surfaces 
and  curves  in  a  space  of  three  dimensions.  Inspired  by  the 
general  activity  of  the  period,  but  following  rather  the  steps  of 
Desargues  and  Pascal,  Carnot  treated  chiefly  the  metrical  rela- 
tions of  figures.  In  particular  he  investigated  these  relations 
as  connected  with  the  theory  of  transversals,  a  theory  whose 
fundamental^  property  of  a  four-rayed  pencil  goes  back  to 
Pappos,  and  which,  though  revived  by  Desargues,  was  set  forth 
for  the  first  time  in  its  general  form  in  Carnot's  Geometric  de 
Position  (1803),  and  supplemented  in  his  Th^orie  des  Trans- 
versales  (1806).  In  these  works  he  introduced  negative  mag- 
nitudes, the  general  quadrilateral  and  quadrangle,  and  numer- 
ous other  generalizations  of  value  to  the  elementary  geometry 
of  to-day.  But  although  Carnot 's  work  was  important  and 
many  details  are  now  commonplace,  neither  the  name  of  the 
theory  nor  the  method  employed  have  endured.  The  present 
fjeometry  of  Position  (Geometric  der  Lage)  has  little  in  com- 
mon with  Carnot's  G^om^trie  de  Position. 

*  On  recent  development  of  graphic  methods  and  the  influence  of  Monge 
upon  this  branch  of  mathematics,  see  Eddy,  H.  T.,  Modern  Graphical  Develop- 
ments, Mathematical  Papers  of  Chicago  Congress  (New  York,  1896),  p.  58. 


560  'history  of  modern  mathematics.  [Chap.  XL 

Projective  Geometry  had  its  origin  somewhat  later  than 
the  period  of  Monge  and  Carnot.  Newton  had  discovered 
that  all  curves  of  the  third  order  can  be  derived  by  central 
projection  from  five  fundamental  types.  But  in  spite  of  this 
fact  the  theory  attracted  so  little  attention  for  over  a  century 
that  its  origin  is  generally  ascribed  to  Poncelet.  A  prisoner 
in  the  Russian  campaign,  confined  at  Saratoff  on  the  Volga 
(1812-14),  "prive,  "  as  he  says,  "  de  toute  espece  de  livres  et 
de  secours,  surtout  distrait  par  les  malheurs  de  ma  patrie  et  les 
miens  propres,"  he  still  had  the  vigor  of  spirit  and  the  leisure 
to  conceive  the  great  work  which  he  published  (1822)  eight 
years  later.  In  this  work  was  first  made  prominent  the  power 
of  central  proje9tion  in  demonstration  and  the  power  of  the 
principle  of  continuity  in  research.  His  leading  idea  was  the 
study  of  projective  properties,  and  as  a  foundation  principle  he 
introduced  the  anharmonic  ratio,  a  concept,  however,  which 
dates  back  to  Pappos  and  which  Desargues  (1639)  had  also 
used.  Mobius,  following  Poncelet,  made  much  use  of  the  an- 
harmonic ratio  in  his  Barycentrische  CalcUl  (1827),  but  under 
the  name  "  Doppelschnitt-Verhaltniss  "  (ratio  bisectionalis),  a 
term  now  in  common  use  under  Steiner's  abbreviated  form 
*'  Doppelverhaltniss."  The  name  "  anharmonic  ratio "  or 
"  function  "  (rapport  anharmonique,  or  fonction  anharmonique) 
is  due  to  Chasles,  and  "  cross-ratio  "  was  coined  by  Clifford. 
The  anharmonic  point  and  line  properties  of  conies  have  been 
further  elaborated  by  Brianchon,  Chasles,  Steiner,  and  von 
Staudt.  To  Poncelet  is  also  due  the  theory  of  "  figures  homo- 
logiques,"  the  perspective  axis  and  perspective  center  (called 
by  Chasles  the  axis  and  center  of  homology),  an  extension  of 
Carnot's  theory  of  transversals,  and  the  "  cordes  ideales  "  of 
conies  which  Pliicker  applied  to  curves  of  all  orders.  He  also 
discovered  what  Salmon  has  called  "  the  circular  points  at  in- 
finity," thus  completing  and  establishing  the  first  great  principle 
of  modern  geometry,  the  principle  of  continuity.  Brianchon 
(1806),  through  his  application  of  Desargues's  theory  of  polajs, 


Art.  16  ]  MODERN    GEOMETRY.  561 

completed  the  foundation  which  Monge  had  begun  for  Ponce- 
let's  (1829)  theory  of  reciprocal  polars. 

Among  the  most  prominent  geometers  contemporary  with 
Poncelet  was  Gergonne,  who  with  more  propriety  might  be 
ranked  as  an  analytic  geometer.  He  first  (1813)  used  the  term 
"  polar"  in  its  modern  geometric  sense,  although  Servois  (1811) 
had  used  the  expression  *'  pole."  He  was  also  the  first  (1825— 
26)  to  grasp  the  idea  that  the  parallelism  which  Maurolycus, 
Snell,  and  Vi^te  had  noticed  is  a  fundamental  principle.  This 
principle  he  stated  and  to  it  he  gave  the  name  which  it  now 
bears,  the  Principle  of  Duality,  the  most  important,  after  that 
of  continuity,  in  modern  geometry.  This  principle  of  geomet- 
ric reciprocation,  the  discovery  of  which  was  also  claimed  by 
Poncelet,  has  been  greatly  elaborated  and  has  found  its  way 
into  modern  algebra  and  elementary  geometry,  and  has  recently 
been  extended  to  mechanics  by  Genese.  Gergonne  was  the 
first  to  use  the  word  "class"  in  describing  a  curve,  explicitly 
defining  class  and  degree  (order)  and  showing  the  duality 
between  the  two.  He  and  Chasles  were  among  the  first  to 
study  scientifically  surfaces  of  higher  order. 

Steiner  (1832)  gave  the  first  complete  discussion  of  the  pro- 
jective relations  between  rows,  pencils,  etc.,  and  laid  the  foun- 
dation for  the  subsequent  development  of  pure  geometry.  He 
practically  closed  the  theory  of  conic  sections,  of  the  corre- 
sponding figures  in  three-dimensional  space  and  of  surfaces  of 
the  second  order,  and  hence  with  him  opens  the  period  of 
special  study  of  curves  and  surfaces  of  higher  order.  His  treat- 
ment of  duality  and  his  application  of  the  theory  of  projective 
pencils  to  the  generation  of  conies  are  masterpieces.  The 
theory  of  polars  of  a  point  in  regard  to  a  curve  had  been 
studied  by  Bobillier  and  by  Grassmann,  but  Steiner  (1848) 
showed  that  this  theory  can  serve  as  the  foundation  for  the 
study  of  plane  curves  independently  of  the  use  of  coordinates, 
and  introduced  those  noteworthy  curves  covariant  to  a  given 
curve  which  now  bear  the  names  of  himself,  Hesse,  and  Cayley, 
This  whole  subject  has  been  extended  by  Grassmann,  Chasles^ 


562  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XL 

Cremona,  and  Jonqui^res.  Steiner  was  the  first  to  make  prom- 
inent (1832)  an  example  of  correspondence  of  a  more  com- 
plicated nature  than  that  of  Poncelet,  Mobius,  Magnus,  and 
Chasles.  His  contributions,  and  those  of  Gudermann,  to  the 
geometry  of  the  sphere  were  also  noteworthy. 

While  Mobius,  Plucker,  and  Steiner  were  at  work  in 
Germany,  Chasles  was  closing  the  geometric  era  opened  in 
France  by  Monge.  His  Apergu  Historique  (1837)  is  a  classic, 
and  did  for  France  what  Salmon's  works  did  for  algebra  and 
geometry  in  England,  popularizing  the  researches  of  earlier 
writers  and  contributing  both  to  the  theory  and  the  nomen- 
clature of  the  subject.  To  him  is  due  the  name  "  homo- 
graphic"  and  the  complete  exposition  of  the  principle  as 
applied  to  plane  and  solid  figures,  a  subject  which  has  received 
attention  in  England  at  the  hands  of  Salmon,  Townsend,  and 
H.  J.  S.  Smith. 

Von  Staudt  began  his  labors  after  Plckuer,  Steiner,  and 
Chasles  had  made  their  greatest  contributions,  but  in  spite  of 
this  seeming  disadvantage  he  surpassed  them  all.  Joining  the 
Steiner  school,  as  opposed  to  that  of  Plucker,  he  became  the 
greatest  exponent  of  pure  synthetic  geometry  of  modern  times. 
He  set  forth  (1847,  1856-60)  a  complete,  pure  geometric  system 
in  which  metrical  geometry  finds  no  place.  Projective  proper- 
ties foreign  to  measurements  are  estabHshed  independently  of 
number  relations,  number  being  drawn  from  geometry  instead 
of  conversely,  and  imaginary  elements  being  systematically 
introduced  from  the  geometric  side.  A  projective  geometry 
based  on  the  group  containing  all  the  real  projective  and  dual- 
istic  transformations,  is  developed,  imaginary  transformations 
being  also  introduced.  Largely  through  his  influence  pure 
geometry  again  became  a  fruitful  field.  Since  his  time  the 
distinction  between  the  metrical  and  projective  theories  has 
been  to  a   great  extent    obliterated,*  the  metrical  properties 

♦Klein,  F.,  Erlangen  Programme  of  1872,  Haskell's  translation,  Bulletio 
of  New  York  Mathematical  Society,  Vol.  II,  p.  215. 


Art.  17.J  ELEMENTARY    GEOMETRY.  563 

being  considered  as  projective  relations  to  a  fundamental  con- 
figuration, the  circle  at  infinity  common  for  all  spheres.  Un- 
fortunately von  Staudt  wrote  in  an  unattractive  style,  and  to 
Reye  is  due  much  of  the  popularity  which  now  attends  the 
subject. 

Cremona  began  his  publications  in  1862.  His  elementary 
work  on  projective  geometry  (1875)  in  Leudesdorf's  translation 
is  familiar  to  English  readers.  His  contributions  to  the  theory 
of  geometric  transformations  are  valuable,  as  also  his  works  on 
plane  curves,  surfaces,  etc. 

In  England  Mulcahy,  but  especially  Townsend  (1863),  and 
Hirst,  a  pupil  of  Steiner's,  opened  the  subject  of  modern 
geometry.  Clifford  did  much  to  make  known  the  German 
theories,  besides  himself  contributing  to  the  study  of  polars 
and  the  general  theory  of  curves. 

Art.  17.    Elementary  Geometry. 

Trigonometry  and  Elementary  Geometry  have  also  been 
affected  by  the  general  mathematical  spirit  of  the  century. 
In  trigonometry  the  general  substitution  of  ratios  for  lines  in 
the  definitions  of  functions  has  simplified  the  treatment,  and 
certain  formulas  have  been  improved  and  others  added.* 
The  convergence  of  trigonometric  series,  the  introduction  of 
the  Fourier  series,  and  the  free  use  of  the  imaginary  have 
already  been  mentioned.  The  definition  of  the  sine  and  cosine 
by  series,  and  the  systematic  development  of  the  theory  on 
this  basis,  have  been  set  forth  by  Cauchy  (1821),  Lobachevsky 
(1833),  and  others.  The  hyperbolic  trigonometry,f  already 
founded  by  Mayer  and  Lambert,  has  been  popularized  and 
further  developed  by  Gudermann  (1830),  Houel,  and  Laisant 
(1871),  and  projective  formulas  and  generalized  figures  have 

*  Todhunter,  I.,  History  of  certain  formulas  of  spherical  trigonometry, 
Philosophical  Magazine,  1873. 

f  Giinther,  S.,  Die  Lehre  von  den  gewohnlichen  und  verallgemeinerten 
Hyperbelfunktionen,  Halle,  1881;  Chrystal,  G.,  Algebra,  Vol.  H,  p.  288. 


564  HISTORY    OF    MODERN    MATHEMATICS.  [ChaP.  XL 

been  introduced,  notably  by  Gudermann,  Mobius,  Poncelet, 
and  Steiner.  Recently  Study  has  investigated  the  formulas 
of  spherical  trigonometry  from  the  standpoint  of  the  modern 
theory  of  functions  and  theory  of  groups,  and  Macfarlane  has 
generalized  the  fundamental  theorem  of  trigonometry  for 
three-dimensional  space. 

Elementary  Geometry  has  been  even  more  affected. 
Among  the  many  contributions  to  the  theory  may  be  men- 
tioned the  following :  That  of  Mobius  on  the  opposite  senses 
of  hues,  angles,  surfaces,  and  soHds ;  the  principle  of  duaHty 
as  given  by  Gergonne  and  Poncelet ;  the  contributions  of  De 
Morgan  to  the  logic  of  the  subject ;  the  theory  of  transversals 
as  worked  out  by  Monge,  Brianchon,  Servois,  Carnot,  Chasles, 
and  others ;  the  theory  of  the  radical  axis,  a  property  dis- 
covered by  the  Arabs,  but  introduced  as  a  definite  concept  by 
Gaultier  (1813)  and  used  by  Steiner  under  the  name  of  "  line 
of  equal  power  "  ;  the  researches  of  Gauss  concerning  inscrip- 
tible  polygons,  adding  the  17-  and  257-gon  to  the  list  below  the 
looo-gon  ;  the  theory  of  stellar  polyhedra  as  worked  out  by 
Cauchy,  Jacobi,  Bertrand,  Cayley,  Mobius,  Wiener,  Hess, 
Hersel,  and  others,  so  that  a  whole  series  of  bodies  have  been 
added  to  the  four  Kepler-Poinsot  regular  solids ;  and  the  re- 
searches of  Muir  on  stellar  polygons.  These  and  many  other 
improvements  now  find  more  or  less  place  in  the  text-books 
of  the  day. 

To  these  must  be  added  the  recent  Geometry  of  the  Tri- 
angle, now  a  prominent  chapter  in  elementary  mathematics. 
Crelle  (18 16)  made  some  investigations  in  this  line,  Feuerbach 
(1822)  soon  after  discovered  the  properties  of  the  Nine-Point 
Circle,  and  Steiner  also  came  across  some  of  the  properties  of 
the  triangle,  but  none  of  these  followed  up  the  investigation. 
Lemoine  *  (1873)  was  the  first  to  take  up  the  subject  in  a  sys- 

*  Smith,  D.  E.,  Biography  of  Lemoine,  American  Mathematical  Monthly, 
Vol.  Ill,  p.  29;  Mackay,  J.  S.,  various  articles  on  modern  geometry  in  Proceed- 
ings Edinburgh  Mathematical  Society,  various  years;  Vigari6,  £.,  G6om6trie  du 
triangle.  Articles  in  recent  numbers  of  Journal  de  Mathematiques  sp6ciales, 
Mathesis,  and  Proceedings  of  the  Association  franjaise  pour  ravancement  des 


Art.  18.]  non-euclidean  geometry.  565 

tematic  way,  and  he  has  contributed  extensively  to  its  de- 
velopment. His  theory  of  "  transformation  continue"  and  his 
'*  geometrographie  "  should  also  be  mentioned.  Brocard's  con- 
tributions to  the  geometry  of  the  triangle  began  in  1877. 
Other  prominent  writers  have  been  Tucker,  Neuberg,  Vigarie, 
Emmerich,  M'Cay,  Longchamps,  and  H.  M.  Taylor.  The 
theory  is  also  greatly  indebted  to  Miller's  work  in  The  Educa- 
tional Times,  and  to  Hoffmann's  Zeitschrift. 

The  study  of  linkages  was  opened  by  Peaucellier  (1864), 
who  gave  the  first  theoretically  exact  method  for  drawing  a 
straight  line.  Kempe  and  Sylvester  have  elaborated  the 
subject. 

In  recent  years  the  ancient  problems  of  trisecting  an  angle, 
doubling  the  cube,  and  squaring  the  circle  have  all  been  settled 
by  the  proof  of  their  insolubility  through  the  use  of  compasses 
and  straight  edge.* 

Art.  18.    Non- Euclidean  Geometry. 

The  Non-Euclidean  Geometry  f  is  a  natural  result  of  the 
futile  attempts  which  had  been  made  from  the  time  of  Proklos 
to  the  opening  of  the  nineteenth  century  to  prove  the  fifth 
postulate  (also   called  the  twelfth  axiom,  and  sometimes  the 

*  Klein,  F.,  Vortrage  iiber  ausgewahlten  Fragen;  Rudio,  F.,  Das  Problem 
von  der  Quadratur  des  Zirkels.  Naturforschende  Gesellschaft  Vierteljahr- 
schrift,  1890;  Archimedes,  Huygens,  Lambert,  Legendre  (Leipzig,  1892). 

f  StJtckel  and  Engel,  Die  Theorie  der  Parallellinien  von  Euklid  bis  auf 
Gauss,  Leipzig,  1895;  Halsted,  G.  B.,  various  contributions:  Bibliography  of 
Hyperspace  and  Non-Euclidean  Geometry,  American  Journal  of  Mathematics, 
Vols.  I,  II;  The  American  Mathematical  Monthly,  Vol.  I;  translations  of  Loba- 
chevsky's  Geometry,  Vasiliev's  address  on  Lobachevsky,  Saccheri's  Geome- 
try, Bolyai's  work  and  his  life;  Non-Euclidean  and  Hyperspaces,  Mathe- 
matical Papers  of  Chicago  Congress,  p.  92.  Loria,  G.,  Die  hauptsachlichsten 
Theorien  der  Geometric,  p.  io6 ;  Karagiannides,  A.,  Die  Nichteuklidische 
Geometric  vom  Altcrthum  bis  zur  Gegenwart,  Berlin,  1893;  McClintock,  E., 
On  the  early  history  of  Non-Euclidean  Geometry,  Bulletin  of  New  York  Mathe- 
matical Society,  Vol.  II,  p.  144;  Poincar6,  Non-Euclidean  Geom,,  Nature, 
45 :  404  ;  Articles  on  Parallels  and  Measurement  in  Encyclopaedia  Britaneica, 
•Qth  edition;  Vasiliev's  address  (German  by  Engel)  also  appears  in  the  Abhand- 
lungen  zur  Geschichte  der  Mathematik,  1895. 


566  HISTORY    OF    MODERN    MATHEMATICS.  [ChAP.  XL 

eleventh  or  thirteenth)  of  Euclid.  The  first  scientific  investi- 
gation of  this  part  of  the  foundation  of  geometry  was  made  by 
Saccheri  (1733),  a  work  which  was  not  looked  upon  as  a  pre- 
cursor of  Lobachevsky,  however,  until  Beltrami  (1889)  called 
attention  to  the  fact.  Lambert  was  the  next  to  question  the 
validity  of  EucHd's  postulate,  in  his  Theorie  der  Parallellinien 
(posthumous,  1786),  the  most  important  of  many  treatises  on 
the  subject  between  the  publication  of  Saccheri's  work  and 
those  of  Lobachevsky  and  Bolyai.  Legendre  also  worked  in 
the  field,  but  failed  to  bring  himself  to  view  the  matter  outside 
the  Euclidean  limitations. 

During  the  closing  years  of  the  eighteenth  century  Kant's* 
doctrine  of  absolute  space,  and  his  assertion  of  the  necessary 
postulates  of  geometry,  were  the  object  of  much  scrutiny  and 
attack.  At  the  same  time  Gauss  was  giving  attention  to  the 
fifth  postulate,  though  on  the  side  of  proving  it.  It  was  at 
one  time  surmised  that  Gauss  was  the  real  founder  of  the  non- 
Euclidean  geometry,  his  influence  being  exerted  on  Loba- 
chevsky through  his  friend  Bartels,  and  on  Johann  Bolyai 
through  the  father  Wolfgang,  who  was  a  fellow  student  of 
Gauss's.  But  it  is  now  certain  that  Gauss. can  lay  no  claim  to 
priority  of  discovery,  although  the  influence  of  himself  and 
of  Kant,  in  a  general  way,  must  have  had  its  effect. 

Bartels  went  to  Kasan  in  1807,  and  Lobachevsky  was  his 
pupil.  The  latter's  lecture  notes  show  that  Bartels  never 
mentioned  the  subject  of  the  fifth  postulate  to  him,  so  that  his 
investigations,  begun  even  before  1823,  were  made  on  his  own 
motion  and  his  results  were  wholly  original.  Early  in  1826 
he  sent  forth  the  principles  of  his  famous  doctrine  of  parallels, 
based  on  the  assumption  that  through  a  given  point  more  than 
one  line  can  be  drawn  which  shall  never  meet  a  given  line 
coplanar  with  it.  The  theory  was  published  in  full  in  1829-30, 
and  he  contributed  to  the  subject,  as  well  as  to  other  branches 
of  mathematics,  until  his  death. 

*  Fink,  E.,  Kant  als  Mathematiker,  Leipzig,  1889. 


Art.  18.]  non-euclidean  geometry.  567 

Johann  Bolyai  received  through  his  father,  Wolfgang,  some 
of  the  inspiration  to  original  research  which  the  latter  had 
received  from  Gauss.  When  only  twenty-one  he  discovered, 
at  about  the  same  time  as  Lobachevsky,  the  principles  of  non- 
Euclidean  geometry,  and  refers  to  them  in  a  letter  of  Novem- 
ber, 1823.  They  were  committed  to  writing  in  1825  and 
pubhshed  in  1832.  Gauss  asserts  in  his  correspondence  with 
Schumacher  (1831-32)  that  he  had  brought  out  a  theory 
along  the  same  lines  as  Lobachevsky  and  Bolyai,  but  the  publi- 
cation of  their  works  seems  to  have  put  an  end  to  his  investi- 
gations. Schweikart  was  also  an  independent  discoverer  of  the 
non-Euclidean  geometry,  as  his  recently  recovered  letters 
show,  but  he  never  published  anything  on  the  subject,  his  work 
on  the  theory  of  parallels  (1807),  like  that  of  his  nephew 
Taurinus  (1825),  showing  no  trace  of  the  Lobachevsky-Bolyai 
idea. 

The  hypothesis  was  slowly  accepted  by  the  mathematical 
world.  Indeed  it  was  about  forty  years  after  its  publication 
that  it  began  to  attract  any  considerable  attention.  Hoiiel 
(1866)  and  Flye  St.  Marie  (1871)  in  France,  Riemann  (1868), 
Helmholtz  (1868),  Frischauf  (1872),  and  Baltzer  (1877)  in  Ger- 
many, Beltrami  (1872)  in  Italy,  de  Tilly  (1879)  *"  Belgium, 
Clifford  in  England,  and  Halsted  (1878)  in  America,  have 
been  among  the  most  active  in  making  the  subject  popular. 
Since  1880  the  theory  may  be  said  to  have  become  generally 
understood  and  accepted  as  legitimate."^ 

Of  all  these  contributions  the  most  noteworthy  from  the 
scientific  standpoint  is  that  of  Riemann.  In  his  Habihtations- 
schrift  (1854)  he  applied  the  methods  of  analytic  geometry  to- 
the  theory,  and  suggested  a  surface  of  negative  curvature, 
which  Beltrami  calls  "pseudo-spherical,"  thus  leaving  EucHd's 
geometry  on  a  surface  of  zero  curvature  midway  between  his; 
own  and  Lobachevsky 's.      He  thus  set  forth  three  kinds  of 

*  Fpr  an  excellent  summary  of  the  results  of  the  hypothesis,  see  an 
article  by  McClintock,  The  Non-Euclidian  Geometry,  Bulletin  of  New  York 
Mathematical  Society,  Vol.  II,  p.   i. 


568  -  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XI. 

geometry,  Bolyai  having  noted  only  two.  These  Klein 
(1871)  has  called  the  elliptic  (Riemann's),  parabolic  (Euclid's), 
and  hyperbolic  (Lobachevsky's). 

Starting  from  this  broader  point  of  view  *  there  have  con- 
tributed to  the  subject  many  of  the  leading  mathematicians  of 
the  last  quarter  of  a  century,  including,  besides  those  already 
named,  Cayley,  Lie,  Klein,  Newcomb,  Pasch,  C.  S.  Peirce, 
Killing,  Fiedler,  Mansion,  and  McClintock.  Cayley 's  contribu- 
tion of  his  "  metrical  geometry  "  was  not  at  once  seen  to  be 
identical  with  that  of  Lobachevsky  and  Bolyai.  It  remained 
for  Klein  (1871)  to  show  this,  thus  simplifying  Cayley's  treat- 
ment and  adding  one  of  the  most  important  results  of  the 
entire  theory.  Cayley's  metrical  formulas  are,  when  the 
Absolute  is  real,  identical  with  those  of  the  hyperbolic  geome- 
try ;  when  it  is  imaginary,  with  the  elliptic  ;  the  limiting  case 
between  the  two  gives  the  parabolic  (Euclidean)  geometry. 
The  question  raised  by  Cayley's  memoir  as  to  how  far  pro- 
jective geometry  can  be  defined  in  terms  of  space  without 
the  introduction  of  distance  had  already  been  discussed  by 
von  Staudt  (1857)  and  has  since  been  treated  by  Klein  (1873) 
and  by  Lindemann  (1876). 

Art.  19.  Bibliography. 
The  following  are  a  few  of  the  general  works  on  the  history 
of  mathematics  in  the  nineteenth  century,  not  already  men- 
tioned in  the  foot-notes.  For  a  complete  bibliography  of  recent 
works  the  reader  should  consult  the  Jahrbuch  uber  die  Fort- 
schritte  der  Mathematik,  the  Bibliotheca  Mathematica,  or  the 
Revue  Semestrielle,  mentioned  below. 

Abhandlungen  zur  Geschichte  der  Mathematik  (Leipzig). 

Ball,  W.  W.  R.,  A  short  account  of  the  history  o^  mathematics 
(London,  1893). 

Ball,  W.  W.   R.,  History  of  the  study  of  mathematics  at  Cam- 
bridge (London,  1889). 

Ball,  W.  W.  R,  Primer  of  the  history  of  mathematics  (London, 
1895). 

*  Klein,  Evanston  Lectures,  Lect.  IX. 


Art.  19.]  bibliography.  569 

Bibliotheca  Mathematica,  G.  Enestrom,  Stockholm.  Quarterly. 
Should  be  consulted  for  bibliography  of  current  articles  and  v/orks 
on  history  of  mathematics. 

Bulletin  des  Sciences  Mathematiques  (Paris,  H^^™®  Partie). 

Cajori,  F.,  History  of  Mathematics  (New  York,  1894). 

Cayley,  A.,  Inaugural  address  before  the  British  Association, 
1883.     Nature,  Vol.  XXVIII,  p.  491. 

Dictionary  of  National  Biography.  London,  not  completed. 
Valuable  on  biographies  of  British  mathematicians. 

D'Ovidio,  Enrico,  Uno  sguardo  alle  origin!  ed  alio  sviluppo  della 
Matematica  Pura  (Torino,  1889). 

Dupin,  Ch.,  Coup  d'oeil  sur  quelques  progres  des  Sciences  mathe- 
matiques, en  France,  1830-35.     Comptes  Rendus,  1835. 

Encyclopaedia  Britannica.  Valuable  biographical  articles  by 
Cayley,  Chrystal,  Clerke,  and  others. 

Fink,  K.,  Geschichte  der  Mathematik  (Tubingen,  1890).  Bib- 
liography on  p.  255. 

Gerhardt,  C.  J.,  Geschichte  der  Mathematik  in  Deutschland 
(Munich,  1877). 

Graf,  J.  H.,  Geschichte  der  Mathematik  und  der  Naturwissen- 
schaften  in  bernischen  Landen  (Bern,  1890).  Also  numerous  bio- 
graphical articles. 

Giinther,  S.,  Vermischte  Untersuchungen  zur  Geschichte  der 
mathematischen  Wissenschaften  (Leipzig,  1876). 

Giinther,  S.,  Ziele  und  Resultate  der  neueren  mathematisch- 
historiscHen  Forschung  (Erlangen,  1876). 

Hagen,  J.  G.,  Synopsis  der  hoheren  Mathematik.  Two  volumes 
(Berlin,  1891-93). 

Hankel,  H.,  Die  Entwickelung  der  Mathematik  in  dem  letzten 
Jahrhundert  (Tiibingen,  1884). 

Hermite,  Ch.,  Discours  prononce  devant  le  president  de  la 
republique  le  5  aout  1889  a  I'inauguration  de  la  nouvelle  Sorbonne. 
Bulletin  des  Sciences  mathematiques,  1890  ;  also  Nature,  Vol.  XLI, 
p.  597.     (History  of  nineteenth-century  mathematics  in  France.) 

Hoefer,  F.,  Histoire  des  mathematiques  (Paris,  1879). 

Isely,  L.,  Essai  sur  I'histoire  des  mathematiques  dans  la  Suisse 
frangaise  (Neuchatel,  1884). 

Jahrbuch  iiber  die  Fortschritte  der  Mathematik  (Berlin,  annu- 
ally, 1868  to  date). 

Marie,  M.,  Histoire  des  sciences  mathematiques  et  physiques. 
Vols.  X,  XI,  XII  (Paris,  1887-88). 

Matthiessen,  L.,  Grundzuge  der  antiken  und  modernen  Algebra 
4er  litteralen  Gleichungen  (Leipzig,  1878). 


570  HISTORY    OF    MODERN    MATHEMATICS.  [Chap.  XI.. 

Newcomb,  S.,  Modern  mathematical  thought.  Bulletin  New 
York  Mathematical  Society,  Vol.  Ill,  p.  95;  Nature,  Vol.  XLIX^ 

P-  325- 

Poggendorff,  J.  C,  Biographisch-literarisches  Handworterbuch 
zur  Geschichte  derexacten  Wissenschaften.  Two  volumes  (Leipzig, 
1863). 

Quetelet,  A.,  Sciences  mathematiques  et  physiques  chez  les 
Beiges  au  commencement  du  XIX^  siecle  (Brussels,  1866). 

Revue  semestrielie  des  publications  mathematiques  redigee  sous- 
les  auspices  de  la  Societe  mathematique  d' Amsterdam.  1893  to  date. 
(Current  periodical  literature.) 

Roberts,  R.  A.,  Modern  mathematics.  Proceedings  of  the  Irish 
Academy,  1888. 

Smith,  H.  J.  S.,  On  the  present  state  and  prospects  of  some 
branches  of  pure  mathematics.  Proceedings  of  London  Mathemat- 
ical Society,  1876;  Nature,  Vol.  XV,  p.  79. 

Sylvester,  J.  J.,  Address  before  the  British  Association.  Nature,, 
Vol.  I,  pp.  237,  261. 

Wolf,  R.,  Handbuch  der  Mathematik.  Two  volumes  (Zurich, 
1872). 

Zeitschrift  fiir  Mathematik  und  Physik.  Historisch-literarische 
Abtheilung.  Leipzig.  The  Abhandlungen  zur  Geschichte  der 
Mathematik  are  supplements. 

For  a  biographical  table  of  mathematicians  see  Fink's  Ge- 
schichte der  Mathematik,  p.  240.  For  the  names  and  positions 
of  living  mathematicians  see  the  Jahrbuch  der  gelehrten  Welt,., 
published  at  Strassburg. 


INDEX. 


671 


INDEX. 


Abelian  functions,  page  545. 
Abel's   quintic    demonstration,    22, 

520. 
Absolute  convergence,  228. 
Absolute,  the,  556. 
Addition  of  vectors,  426,  443. 
Adjustment  of  observations,  469. 
Algebraic  equations,  1-32. 
Alternants,  527. 
Alternating  current,  153 

electromotiveforce,  440, 
442. 
Analytic  geometry,  552. 
Anharmonic  ratio,  560. 
Anti-hyperbolic  functions,  116. 
Approximation  of  roots,  3,  12,  519. 
Arched  catenary,  146. 
Array  of  determinant,  36. 
Associative  law,  392. 
Ausdehnungslehre,  374-424,  517. 
Automorphic  functions,  549. 
Average  error,  493. 
Axial  projection,  72. 

Bernoulli's  theorem,  482. 
Bibliography,  508,  568. 
Binary  forms,  513,  531. 
Binomial  equations,  16,  522. 

factors,  46. 

theorem.  462. 
Bessel's  functions,  169,  183,  213,  215, 

220,  221,  224,  345,  549. 

Branch  of  a  function,  251. 

point,  252. 

Canonical  forms,  531. 


Calculus,  531. 

of  variations,  533. 
Catenary,  14,  145,  327. 

of  uniform  strength,  147* 
Cauchy's  expansion,  52. 

theorem,  262,  264. 
Center  and  diameter,  94. 
Central  axis,  456. 
Centroids,  381,  386. 
Chance,  games  of,  467. 
Characteristic  ratios  in  conies,  no. 
Clairaut's  equation,  324. 
Coaxial  quaternions,  439. 
Cofactors  in  determinants,  47,  62. 
Combinations,  473. 
Combinatory  multiplication,  392. 
Commutative  law,  392. 
Complete    integral,    309,    333,    359,, 

362. 
Complement,  the,  399,  416. 
Complementary  function,  399,  416. 
Composition  of  quantities,  453. 
of  rotations,  463. 
of  vectors,  428. 
Complex  hyperbolic  functions,  140. 

integrals,  261. 

numbers,  138,  515. 

roots.  31. 

variable,  226-302. 
Concurrent  events,  479. 
Conduction  of  heat,  183,  219. 
Conformal  representation,  236,  238, 

244. 
Congruences,  512. 
Conic  ranges,  91. 
Conies,  points  on,  107. 


57g 


HIGHER    MATHEMATICS. 


Conies,  projective  generation  of ,  86, 

561. 

sectors  of,  no. 

triangles  of,  no. 
Conjugate  functions,  245. 
Cosine  series,  192. 
Consistence  of  equations,  58. 
Continued  fractions,  515. 
Continuity  of  functions,  230. 
Convergence,  of  series.  540. 
uniform,  274. 
Conversion     hyperbolic     formulas, 

118. 
Cooling  of  iron  plate,  174. 
Coplanar  vectors,  426,  432. 
Correlation,  74. 
Correspondence  in  conies,  107. 
Cosine  series,  192. 
Covariants,  529. 
Critical  lines,  298. 

points,  250,  278. 

regions,  298. 
Cross  ratio,  104. 
Cubic  equations,  17. 
surfaces,  556. 
Curves  of  second  degree,  82. 
Curvilinear  integrals,  267. 
Cusp  locus,  319. 
Cut,  251. 
Cutting,  72. 
Cylinder,  floating,  13. 

Deficiency  curves,  548. 
De  Moivre's  formula,  515, 

quintic,  23. 
Derivative  equation,  9. 
Derivatives   of   complex    functions, 
233- 
gudermanians,  130. 
hyperbolic   func- 
tions, 120,  122. 
Descriptive  geometry,  554. 
Determinant  array,  36. 
Determinants,  33-69,  408,  526. 
Development  of  determinants, 49, 52. 
Differential  calculus,  531. 

equations,  172,  303-373, 
535. 


Differentiation,  54,  120,  322. 
Direct  probabilities,  476. 
Dirichlet's  conditions,  198. 
Discontinuous  groups,  526. 
Discriminant,  319. 
Distributive  law,  392. 
Doubly  ruled  surfaces,  102. 
Duality  of  plane  and  point,  96. 

principle  of,  74. 
Dynamo,  445. 

Elastic  catenary,  148. 
Elementary  functions,  229. 
geometry,  564. 
Electric  charges,  177. 

currents,   153,  305,  440,  452. 
motor,  445. 
Elements  at  infinity,  72. 
of  a  group,  34. 
of  roots,  15. 
Eliminant,  59,  61. 
Elimination,  523. 
Ellipse,  functional  relations,  ill. 
Elliptic  functions,  23,  107,  291,  522, 
544- 
modular  functions,  549. 
Equation  of  energy,  331. 
Equations,     differential,      169,     172, 

303-373.  535. 

solution  of,  1-32. 

theory  of,  519. 
Equilibrium  of  forces,  404. 
Equipotential  curves,  248. 
Energy,  equation  of,  332. 
Envelopes,  317. 
Error  function,  492. 
Errors,  theory  of,  467-507,  550. 
Essential  singularity,  253. 
Eulerian  integrals,  532. 
Exact  differential  equations,  308. 
Expansion  of  hyperbolic  functions, 

123,  125. 
Exponential  expressions,  124. 

functions,  229. 

theorem,  460. 

First  integrals,  332,  333. 
Flexure  and  tension,  151. 
Flotation  of  bodies,  13. 


INDEX. 


57a 


Fluid  motion,  246. 
Flux  across  a  curve,  247. 
magnetic,  445,  452. 
Forces,  equilibrium  of,  404. 

resultant  of,  398,  430. 
Forms,  530. 

Fourier's  series,  194,  196,  273,  542. 
Fractions,  continued,  514. 
Functions,  Bessel's,  169-220,  549. 

determinantal,  37. 

elliptic,  107,  522,  544. 

harmonic,  169-225. 

hyperbolic,  107-168. 

of  a   complex    variable, 
226-302. 

symbols  for,  543. 

theory  of,  543, 

with  n  values,  300. 

Galois's  group  theory,  525. 
Games  of  chance,  467. 
Gamma  functions,  488,  533. 
General  integrals,  359. 
Geometric  applications,  325. 
elements,  70. 
multiplication,  390. 
representation,  227,  305. 
Geometry,  analytic,  552. 

descriptive,  558. 
elementary.  563. 
modern,  558. 
non-Euclidean,  104,  565. 
projective,  76-106,  560. 
Graphic  representations,  232. 

solution  of  equations,  3. 
Graphs  of  equations,  9. 

of  hyperbolic  functions,  132. 
of  laws  of  errors,  492,  496, 
500,  503. 
Grassmann's   space    analysis,    374- 

424. 
Groups,  34,  524. 
Gudermanian  angle,  ^29. 

function,  128. 
tables  of,  168. 

Harmonic  analysis,  169. 
elements,  77. 
functions,  169-225. 


Heat  conduction,  174,  183,  215,  219. 

Hessian  curve,  555. 

History    of    modern    mathematics,. 

508-570. 
Holomorphic  function,  256. 
Homogeneous  differential  equations,, 
172,  311,  342. 
linear  equations,  60. 
Horner's  method,  2,  12,  519. 
Howe  truss  strut  problem,  21. 
Hyperbola,      functional     relations, 

III. 
Hyperbolic  functions,  107-168. 

paraboloid.  99. 
Hyperboloid  of  one  nappe,  100. 
Hyperdeterminants,  525. 
Hyperelliptic  functions,  546. 
Hypergeometric  series,  348,  540, 

Icosahedron  equation,  521. 
Imaginaries,  141,  227,  516. 
Infinitesimal  calculus,  534. 
Infinite  series,  539. 
Infinity,  elements  at,  73. 

point  at,  256. 
Inner  products,  401. 
Integral  calculus,  531. 
Integral,  complex,  261. 

curvilinear,  267. 

hyperbolic,  135. 

one-valued  function,  284. 

probability,  483. 
Integrals,  257,  303-373- 
Intermediate    differential  equation, 

366. 
Interpolated  values,  497. 
Interpolation  formulas,  541. 
Invariants,  531. 
Inverse  probabilities,  484. 
Inversions  in  permutations,  35. 
Involution,  88. 
Irrational  numbers,  513. 
Irrotational  motion,  248. 

Jacobians,  357. 

Journals,  mathematical,  509. 

Kern  curve,  88. 


574 


HIGHER    MATHEMATICS. 


Lagrange's  equation,  356. 

lines,  358. 

resolvent,  15,  520. 

series,  549. 
Lam  's  functions,  221. 
Laplace's  equation,  170,  203. 
Laurent's  series,  271. 
Laws  of  error,  491. 
Least  squares,  470,  550. 
Legendre's  equation,  179. 
Limiting  values,  119. 
Linear  algebraic  equations,  56. 

differential    equations,     172, 
312,  336,  338,  368. 
Literature,  436,  569. 
Locus,  317,  319,  321. 
Logarithmic  branch  point,  252. 

discontinuity,  252. 

expressions,  127. 

functions,  229. 

solution   of  equations, 
32. 

tables,  497,  499.  503. 
Loxpdrome,  150. 

Maclaurin's  configuration,  35. 

series,  271. 
Magnetic  flux,  445,  449. 
Magnification,  237. 
Map  drawing,  237. 
Mathematical  bibliography,  568. 

history,  508-570. 

periodicals,  509. 
Mathematicians,  living,  570. 
Matrix,  60. 
McClintock's  method  for  equations, 

29. 
Mean  error,  40  J. 
Mercator's  projection,  151,  245. 
Mereomorphic  function,  256. 
Metrical  geometry,  104,  568. 
Minimum  surfaces,  557. 
Minor  determinants,  47, 
Mittag-Leffler's  theorem,  292,  548. 
Models,  557. 
Modern  geometry,  558. 

mathematics,  508-570. 
Modulus  of  complex  variable,  227. 


Modulus  of  integral,  259. 
Monge's  equations,  367. 
Monogenic  function,  233,  235. 
Multiplication,  geometric,  390,  457, 
theorem,  65. 

Newton's  approximation  rule,  6. 
Node  locus,  321. 

Non-Euclidean  geometry,  104,  565. 
Notations  for  determinants,  39. 

for  functions,  543. 

for  vectors,  427. 
Numbers,  projective  definition,  104. 

theory  of,  511,  513. 
Numerical  equations,  10,  519. 

Observations,  errors  of,  550. 
Omega  functions,  547. 
One-valued  functions,  226,  278. 
Orders  of  determinants,  40. 
Orthogonal  trajectories,  326. 

Parallel  lines,  43,  566. 
Partial  derivatives,  245. 

differential    equations,    355, 
365,  368,  535. 
Particular  integral,  337. 
Pencils,  71. 
Periodicals,  509. 

Periodic  functions,  169-225,  547. 
Permutations,  34,  471. 
Physics,  534. 

Plane  and  point  duality,  96. 
Plane  perspective,  76. 

sects,  412. 

vectors,  413. 
Planes,  sum  of,  413. 
Planimetric  products,  390,  392. 
Plucker's  equations,  554,  556. 
Point  analysis,  374-424. 

at  infinity,  256. 
Points,  sum  of,  375. 
Pole  and  polar,  87,  559,  561. 
Polygons,  565.. 
Polygrams,  74. 
Polyhedra,  564. 
Polystims,  74. 
Potential,  177,  534. 
Primal  forms,  70. 


INDEX. 


575 


Prime  numbers,  512. 
Primitive  of  differential,  307. 
Probable  error,  493. 
Probabilities,  467-507,  550. 
Product  of  arrays,  68. 

of  determinants,  67. 

of  points,  392,  410. 

of  sects,  394,  411. 

of    vectors,    394,    432,    444, 

449- 
of  versors,  459. 
Products,  planimetric,  390,  392. 

stereometric,  390,  410. 
Projecting,  72. 
Projective  conic  ranges,  91. 

geometry,  70-106,  560. 
Projectivity,  80. 

Quadrantal  versors,  457. 
Quadratic  equations,  16. 
Quadric  surfaces,  98. 
Quantics,  528. 
Quantity,  complex,  515. 
Quartic  equations,  19. 
Quaternions,  425-466,  517. 
Quintic  equations,  21,  520. 

Raising  the  order,  55. 
Reciprocal  of  vectors,  435. 

determinants,  69. 
Reciprocity,  512. 
Rectangle    inscribed    in    rectangle, 

20,  25. 
Reference  systems,  386. 
Regions,  critical,  298. 
Regula  falsi,  5. 
Removal  of  terms,  11,  22,  520. 
Representation,  conformal,  236,238, 
244. 
geometric,  305. 
graphic,  232. 
Residues,  282,  512. 
Resolvent  equations,  17. 

sextic,  23,  520. 
Resultant  error,  494. 
Resultant  of  equations,  59,  61,  523. 

of  forces,  398. 

of  vectors,  429. 


Roots  of  Bessel's  functions,  225. 

of  equations,  1-32,  520. 

of  unity,  15. 
Rotations,  463. 
Rows  and  columns,  42. 
Ruled  quadric  surfaces,  98. 
Rules  for  versors,  457. 

Sarrus's  rule,  40. 

Scalar  products,  433,  446. 

quantities,  375,  433. 
Schools  of  mathematics,  509. 
Second-degree  curves,  82. 
Semi-convergent  series,  534. 
Separation  of  roots,  8. 

of  variables,  304. 
Series,  convergence  of,  274. 

for  roots,  27,  30. 

Fourier's,  273. 

infinite,  228,  342,  537. 

Laurent's,  271' 

Maclaurin's,  271. 

Taylor's,  269. 

trigonometric,  174,  542. 
Sextic  resolvent,  17,  520. 
Simultaneous  differential  equations, 

327. 
Sine  series,  188,  542. 
Singular  solutions,  317,  320,  537. 
Singularity,  essentia],  253. 
Solution  of  equations.  1-32.  519. 
linear  equations,  56. 
Space  analysis,  374-424,  425. 
Sphere,    conformal    representation, 
244. 
depth  of  immersion,  13,  15. 
Spherical  harmonics,  169,  213. 

trigonometry,  418,  457. 
versors,  458. 
Stereometric  products,  391,  410. 
Stirling's  theorem,  482. 
Stream  function,  247. 
Sturm's  theorem,  2,  8,  12,  519. 
Substitutions,  524. 
Sum  and  difference  formulas,  116. 
of  points,  375. 
Surfaces,  ruled,  98. 
Surveying  problems,  8a 


670 


HIGHER    MATHEMATICS. 


Sylvester's  method  of  elimination, 

63. 
Symbolic  methods,  533. 
Symmetric  functions,  523. 
Synectic  function,  256. 
Synthetic  geometry,  552. 
Systems  of  curves,  306,  325,  333. 

of    differential     equations, 
349- 

Tables  of  Bessel's  functions,  224. 

of  combinations,  475. 

of  gudermanians,  168. 

of  hyperbolic  functions,  160- 
168. 

of  permutations,  471. 

of  probability  integral,  484. 

of  roots  of  Bessel's  functions, 
225. 

of  surface  zonal  harmonics, 
222. 

of  values  oi  Jo{xi),  225. 
Tabular  values,  494,  503. 
Tac  locus,  318. 
Taylor's  series,  269. 
Tension  and  flexure,  151. 

in  catenary,  14,  146. 
Ternary  forms,  509. 
Tetrahedra,  412,  422. 
Theory  of  errors,  467-507. 

of  functions,  226,  543. 

of  numbers,  511. 
Theta  functions,  546. 


Torque,  455. 
Tractor  y,  149. 
Trajectories,  325. 
Transcendent  equations,  I-15. 

functions,  538. 

numbers,  513. 
Triangle,  geometry  of,  564. 
Trigonometric  series,  174,  200. 

solution  of  equations, 
24. 
Trigonometry,  559. 
Typical  errors,  493. 

Uniform  convergence,  274,  541. 
Uniform  strength,  catenary  of,  147. 

Variations,  calculus  of,  533. 
Vector  analysis.  425-466. 

products,  434,  447. 

quantities,  138,  374-466,  518. 
Versors,  457. 

Water  pipe,  13. 
Weierstrass's  ^-function,  297. 

theorem,  287. 
Weighted  points,  378. 
Whist,  game  of,  477. 

Zero  determinants,  62. 

formulas,  51. 
Zonal  harmonics,  169,  177,  202,  205, 
208,  212,  222. 


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Dana's  Elementary  Mechanics 12mo,  1  50 

Wood's          •'                 "           12mo,  125 

"               "                 "           Supplement  and  Key 125 

Analytical  Mechanics Svo,  3  00 

Michie's  Analytical  Mechanics Svo,  4  00 

Merriman's  Mechanics  of  Materials Svo,  4  00 

Church's  Mechanics  of  Engineering Svo,  6  00 

"        Notes  and  Examples  in  Mechanics Svo,  2  00 

Mosely's  Mechanical  Engineering.     (Mahan.) Svo,  5  00 

Weisbach's    Mechanics    of   Engineering.     Vol.    III.,   Part  I., 

Sec.  L     (Klein.)...  Svo,  5  00- 

Weisbach's  Mechanics    of  Engineering.     Vol.   III.,    Part  I. 

Sec.IL     (Klein.) Svo,  5  00 

Weisbach's  Hydraulics  and  Hydraulic  Motors.    (Du  Bois.)..Svo,  5  00^ 

Steam  Engines.     (Du  Bois.) Svo,  5  00 

Lanza's  Applied  Mechanics Svo,  7  50^ 

11 


<Jrehore's  Mechanics  of  the  Girder 8vo,  $5  00 

lllacCord's  Kinematics 8vo,  5  00 

Thurston's  Friction  and  Lost  Work 8vo,  3  00 

"         The  Animal  as  a  Machine 12mo,  1  00 

Hall's  Car  Lubrication 12mo,  1  00 

Warren's  Machine  Construction 3  vols. ,  Svo,  7  50 

"Chordal's  Letters  to  Mechanics 12mo,  2  00 

The  Lathe  and  Its  Uses Svo,  6  00 

■Cromwell's  Toothed  Gearing 12mo,  1  50 

Belts  and  Pulleys 12mo,  150 

Du  Bois's  Mechanics.     Vol.  I.,  Kinematics Svo,  3  50 

Vol.  IL,  Statics Svo,  4  00 

Vol.  IIL,  Kinetics Svo,  3  50 

Dredge's     Trans.     Exhibits     Building,      World     Exposition, 

4to,  half  morocco,  15  00 

Flather's  Dynamometers 12mo,  2  00 

Rope  Driving 12mo,  2  00 

Hichards's  Compressed  Air .'.'". 12mo,  1  50 

Smith's  Press-working  of  Metals Svo,  3  00 

Holly's  Saw  Filing ISmo,  75 

Fitzgerald's  Boston  Machinist ISmo,  1  00 

JBaldwiu's  Steam  Heating  for  Buildings 12mo,  2  50 

Metcalfe's  Cost  of  Manufactures Svo,  5  00 

JBen jamin's  Wrinkles  and  Recipes .' 12mo,  2  00 

Dingey's  Machinery  Pattern  Making 12mo,  2  00 

METALLURGY. 

Iron— Gold— Silver — Alloys,  Etc. 

iEgleston's  Metallurgy  of  Silver Svo,  7  50 

Gold  and  Mercury Svo,  7  50 

**         Weights  and  Measures,  Tables ISmo,  75 

**          Catalogue  of  Minerals Svo,  2  50 

O'Driscoll's  Treatment  of  Gold  Ores. . . .  ;\l  i'"/i  f. Svo,  2  00 

*  Kerl's  Metallurgy— Copper  and  Iron. .....'!'.'!. Svo,  15  00 

*  '*           "               Steel,  Fuel,  etc Svo,  15  00 

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Thurston's  Iron  and  Steel 8vo,  $3  50 

Alloys 8vo,  3  50 

Troilius's  Chemistry  of  Iron 8vo,  3  OO 

Kiinbardt's  Ore  Dressing  in  Europe Svo,  1  50 

Weyrauch's  Strength  of  Iron  and  Steel.    (Du  Bois.) Svo,  1  50 

Beardslee  and  Kent's  Strength  of  Wrought  Iron Svo,  1  50 

Compton's  First  Lessons  in  Metal  Working 13mo,  1  50 

West's  American  Foundry  Practice 13mo,  3  50 

"     Moulder's  Text-book 13mo,  3  50 


MINERALOGY  AND  MINING. 

Mine  "Accidents — Ventilation— Ore  Dressing,  Etc. 

Dana's  Descriptive  Mineralogy,     (E.  S.) Svo,  half  morocco, 

"      Mineralogy  and  Petrography.     (J.  D.) 13mo, 

"      Text-book  of  Mineralogy.    (E.  S.) Svo, 

"      Minerals  and  How  to  Study  Them.     (E.  S.) 13mo, 

"      American  Localities  of  Minerals Svo, 

Brush  and  Dana's  Determinative  Mineralogy. Svo, 

Rosenbusch's    Microscopical    Physiography  of    Minerals    and 

Rocks.     (Iddiugs.) Svo, 

Hussak's  Rock  forming  Minerals.     (Smith.) Svo, 

Williams's  Lithology Svo, 

Chester's  Catalogue  of  Minerals Svo, 

"       Dictionary  of  the  Names  of  Minerals Svo, 

Egleston's  Catalogue  of  Minerals  and  Synonyms Svo, 

Goodyear's  Coal  Mines  of  the  Western  Coast 13mo, 

Kunhardt's  Ore  Dressing  in  Europe Svo, 

Sawyer's  Accidents  in  Mines Svo, 

Wilson's  Mine  Ventilation 16mo, 

Boyd's  Resources  of  South  Western  Virginia Svo, 

Map  of  South  Western  Virginia Pocket-book  form, 

Stockbridge's  Rocks  and  Soils Svo, 

Eissler's  Explosives — Nitroglycerine  and  Dynamite Svo, 

13 


.3  50 

3  00 

3  50 

1  50 

1  OO 

3  50 

5  00 

3  OO 

3  00 

1  35 

3  00 

3  50 

3  50 

1  50 

7  00 

1  35 

3  00 

3  00 

3  50 

4  00 

*Drinker's  Tunuelling,  Explosives,  Compounds,  and  Rock  Drills. 

l4to,  half  morocco,  |25  00 

Beard's  Ventilation  of  Mines . . 12mo,  2  50 

Ihlseng's  Manual  of  Mining j,^^  Mrt^feSMCI* '  *  •  •^^^'  '*  ^^ 

STEAM  AND  ELECTRICAL  ENGINES,  BOILERS,  Etc. 
Stationary — Marine— Locomotive — Gas  Engines,  Etc. 

Weisbach's  Steam  Engine.     (Du  Bois.)..  .^.. .,,,.. 8vo,  5  00 

'Thurston's  Engine  and  Boiler  Trials •  rrrv ^^o-  ^  ^^ 

Philosophy  of  the  Steam  Engine 12mo,  75 

Stationary  Steam  Engines 12mo,  1  50 

Boiler  Explosion.. 12mo,  150 

'    Steam-boiler  Construction  and  Operation 8vo, 

Reflection  on  the  Motive  Power  of  Heat.    (Caruot.) 

12mo,  2  00 
Thurston's  Manual  of  the  Steam  Engine.     Part  I.,  Structure 

and  Theory 8vo,  7  50 

Thurston's  Manual  of  the  Steam  Engine.     Part  IL,   Design,  . 

Construction,  and  Operation 8vo,  7  50 

2  parts,  12  00 

R5ntgen's  Thermodynamics.     (Du  Bois.) 8vo,  5  00 

Peabody's  Thermodynamics  of  the  Steam  Engine 8vo,  5  00 

"         Valve  Gears  for  the  Steam-Engine 8vo,  2  50 

Tables  of  Saturated  Steam 8vo,  1  00 

Wood's  Thermodynamics,  Heat  Motors,  etc 8vo,  4  00 

Pupin  and  Osterberg's  Thermodynamics 12mo,  1  25 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  1  50 

Reagan's  Steam  and  Electrical  Locomotives 12mo,  2  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Whitham's  Steam-engine  Design 8vo,  6  00 

Constructive  Steam  Engineering 8vo,  10  00 

Hemen way's  Indicator  Practice 12mo,  2  00 

Pray's  Twenty  Years  with  the  Indicator Royal  8vo,  2  50 

Spangler's  Valve  Gears 8vo,  2  50 

*  Maw's  Marine  Engines Folio,  half  morocco,  18  00 

"Trowbridge's  Stationary  Steam  Engines 4to,  boards,  2  50 

14 


Pord's  Boiler  Making  for  Boiler  Makers 18mo,  $1  00 

Wilson's  Steam  Boilers.     (Flather. ) 12mo,  2  50 

Baldwin's  Steam  Heating  for  Buildings 12mo,  2  50 

Hoadley's  Warm-blast  Furnace 8vo,  1  50 

Sinclair's  Locomotive  Running 12mo,  2  00 

•Clerk's  Gas  Engine 12mo, 

TABLES,  WEIGHTS,  AND  MEASURES. 

PoR  Engineers,  Mechanics,  Actuaries— Metric  Tables,  Etc. 

'Crandall's  Railway  and  Earthwork  Tables 8vo,  1  50 

Johnson's  Stadia  and  Earthwork  Tables Svo,  1  25 

Bixby's  Graphical  Computing  Tables Sheet,  25 

•€ompton's  Logarithms 12mo,  1  50 

Ludlow's  Logarithmic  and  Other  Tables.     (Bass.) 12rao,  2  00* 

Thurston's  Conversion  Tables ...   . .  .Svo,  1  00 

Egleston's  Weights  and  Measures ISmo,  75 

Totten's  Metrology Svo,  2  50 

Fisher's  Table  of  Cubic  Yards. Cardboard,  25 

Hudson's  Excavation  Tables.     Vol.  H Svo,  1  00 

VENTILATION. 

Steam  Heating — House  Inspection — Mine  Ventilation. 

beard's  Ventilation  of  Mines 12mo,  2  50 

Baldwin's  Steam  Heating 12mo,  2  50 

Reid's  Ventilation  of  American  Dwellings 12mo,  1  50 

Mott's  The  Air  We  Breathe,  and  Ventilation 16mo,  1  00 

Gerhard's  Sanitary  House  Inspection Square  16mo,  1  00 

Wilson's  Mine  Ventilation 16mo,  1  25 

Carpenter's  Heating  and  Ventilating  of  Buildings Svo,  3  00 

niSCELLANEOUS  PUBLICATIONSo 

Aicott's  Gems,  Sentiment,  Language. . . .  ..■.•;  .•.•.-.■. .  .Gilt  edges,  5  00 

Bailey's  The  New  Tale  of  a  Tub QX^4l?^9VX. . .  .Svo,  75 

Ballard's  Solution  of  the  Pyramid  Problem Svo,  1  50 

jBarnard's  The  Metrological  System  of  the  Great  Pyramid.  .Svo,  1  50 

15 


*  Wiley's  Yosemite,  Alaska,  and  Yellowstone 4to,  |3  OO 

Emmon's  Geological  Guide-book  of  the  Rocky  Mountains.  .8vo,  1  50 

Ferrel's  Treatise  on  the  Winds 8vo,  4  OO 

Perkins's  Cornell  University Oblong  4to,  1  50 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute 8vo,  3  OO 

Mott's  The  Fallacy  of  the  Present  Theory  of  Sound.  .8q.  16mo,  1  OO 
Rotherham's    The    New    Testament    Critically  Emphathized. 

12mo,  1  50 

Totten's  An  Important  Question  in  Metrology Svo,  3  50 

Whitehouse*s  Lake  Moeris Paper,  25- 

HEBREW  AND  CHALDEE  TEXT-BOOKS. 

For  Schoolb  and  Theological  Seminaries. 

Gesenius's  Hebrew  and   Chaldee  Lexicon  to  Old   Testament. 

(Tregelles.). ... Small  4to,  half  morocco,  5  OO 

Green's  Grammar  of  the  Hebrew  Language  (New  Edition). Svo,  3  00 

*•       Elementary  Hebrew  Grammar. . .   12mo,  1  25^ 

"       Hebrew  Chrestomathy Svo,  2  OO 

Letteris's    Hebrew  Bible  (Massoretic  Notes  in  English). 

Svo,  arabesque,  2  25 
Luzzato's  Grammar  of  the  Biblical  Chaldaic  Language  and  the 

Talmud  Babli  Idioms 12mo,  1  50 

MEDICAL. 

Bull's  Maternal  Management  in  Health  and  Disease 12mo,  1  00 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food. 

Large  mounted  chart,  1  25- 

Steel's  Treatise  on  the  Diseases  of  the  Ox Svo,  6  OO 

Treatise  on  the  Diseases  of  the  Dog Svo,  3  50 

Worcester's  Small  Hospitals— Establishment  and  Maintenance, 
including  Atkinson's  Suggestions  for  Hospital  Archi- 
tecture  12mo,  1  25 

Hammarsten's  Physiological  Chemistry.   (Mandel.) Svo,  4  00 


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